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I've spent a few days trying to figure this out and looking up tutorials, but everything I've found so far seems like it's close to what I need but don't give the results I need.
I have a device that produces a single letter, A-F. For simplicity's sake, you can think of it like a die with letters. It will always produce one and only one letter each time it is used. However it has one major difference: each letter can have a differing known probability of being picked:
A: 25%
B: 5%
C: 20%
D: 15%
E: 20%
F: 15%
These probabilities remain constant throughout all attempts.
Additionally, I have a specific combination I must accrue before I am "successful":
As needed: 1
Bs needed: 3
Cs needed: 0
Ds needed: 1
Es needed: 2
Fs needed: 3
I need find the average number of letter picks (i.e. rolls/trials/attempts) that have to happen for this combination of letters to be accrued. It's completely fine for any individual outcome to have more than the required number of letters, but success is only counted when each letter has been chosen at least its minimum amount of times.
I've looked at plenty of tutorials for multinomial probability distribution and similar things, but I haven't found anything that explains how to find average number of trials for a scenario like this. Please kindly explain answers clearly as I'm not a wiz with statistics.
In addition to Severin's answer that logically looks good to me but might be costly to evaluate (i.e. infinite sum of factorials).
Let me provide some intuition that should give a good approximation.
Considering each category at a time. Refer this math stackexchange question/ answer. Expected number of tosses in which you would get the k number of successes for each category (i) can be calculated as k(i)/ P(i):
Given,
p(A): 25% ; Expected number of tosses to get 1 A = 1/ 0.25 = 4
p(B): 5% ; Expected number of tosses to get 3 B's = 3/ 0.05 = 60
p(C): 20% ; Expected number of tosses to get 0 C = 0/ 0.20 = 0
p(D): 15% ; Expected number of tosses to get 1 D = 1/ 0.15 = 6.67 ~ 7
p(E): 20% ; Expected number of tosses to get 2 E's = 2/ 0.20 = 10
p(F): 15% ; Expected number of tosses to get 3 F's = 3/ 0.15 = 20
you get an idea that getting 3 B's is your bottleneck, you can expect on average 60 tosses for your scenario to play out.
Well, minimum number of throws is 10. Average would be infinite sum
A=10•P(done in 10)+11•P(done in 11)+12•P(done in 12) + ...
For P(done in 10) we could use multinomial
P(10)=Pm(1,3,0,1,2,3|probs), where probs=[.25, .05, .20, .15, .20, .15]
For P(11) you have one more throw which you could distribute like this
P(11)=Pm(2,3,0,1,2,3|probs)+Pm(1,4,0,1,2,3|probs)+Pm(1,3,0,2,2,3|probs)+
Pm(1,3,0,1,3,3|probs)+Pm(1,3,0,1,2,4|probs)
For P(12) you have to distribute 2 more throws. Note, that there are combinations of throws which are impossible to get, like Pm(2,3,0,2,2,3|probs), because you have to stop earlier
And so on and so forth
Your process can be described as a Markov chain with a finite number of states, and an absorbing state.
The number of steps before reaching the absorbing state is called the hitting time. The expected hitting time can be calculated easily from the transition matrix of the Markov chain.
Enumerate all possible states (a, b, c, d, e, f). Consider only a finite number of states, because "b >= 3" is effectively the same as "b = 3", etc. The total number of states is (1+1)*(3+1)*(0+1)*(2+1)*(3+1) = 192.
Make sure that in your enumeration, starting state (0, 0, 0, 0, 0, 0) comes first, with index 0, and absorbing state (1, 3, 0, 1, 2, 3) comes last.
Build the transition matrix P. It's a square matrix with one row and column per state. Entry P[i, j] in the matrix gives the probability of going from state i to state j when rolling a die. There should be at most 6 non-zero entries per row.
For example, if i is the index of state (1, 0, 0, 1, 2, 2) and j the index of state (1, 1, 0, 1, 2, 2), then P[i, j] = probability of rolling face B = 0.05. Another example: if i is the index of state (1,3,0,0,0,0), then P[i,i] = probability of rolling A, B or C = 0.25+0.05+0.2 = 0.5.
Call Q the square matrix obtained by removing the last row and last column of P.
Call I the identity matrix of the same dimensions as Q.
Compute matrix M = (I - Q)^-1, where ^-1 is matrix inversion.
In matrix M, the entry M[i, j] is the expected number of times that state j will be reached before the absorbing state, when starting from state i.
Since our experiment starts in state 0, we're particularly interested in row 0 of matrix M.
The sum of row 0 of matrix M is the expected total number of states reached before the absorbing state. That is exactly the answer we seek: the number of steps to reach the absorbing state.
To understand why this works, you should read a course on Markov chains! Perhaps this one: James Norris' course notes on Markov chains. The chapter about "hitting times" (which is the name for the number of steps before reaching target state) is chapter 1.3.
Below, an implementation in python.
from itertools import product, accumulate
from operator import mul
from math import prod
import numpy as np
dice_weights = [0.25, 0.05, 0.2, 0.15, 0.2, 0.15]
targets = [1, 3, 0, 1, 2, 3]
def get_expected_n_trials(targets, dice_weights):
states = list(product(*(range(n+1) for n in targets)))
base = list(accumulate([n+1 for n in targets[:0:-1]], mul, initial=1))[::-1]
lookup = dict(map(reversed, enumerate(states)))
P = np.zeros((len(states), len(states)))
for i, s in enumerate(states):
a,b,c,d,e,f = s
for f, p in enumerate(dice_weights):
#j = index of state reached from state i when rolling face f
j = i + base[f] * (s[f] < targets[f])
j1 = lookup[s[:f] + (min(s[f]+1, targets[f]),) + s[f+1:]]
if (j != j1):
print(i, s, f, ' --> ' , j, j1)
assert(j == j1)
P[i,j] += p
Q = P[:-1, :-1]
I = np.identity(len(states)-1)
M = np.linalg.inv(I - Q)
return M[0,:].sum()
print(get_expected_n_trials(targets, dice_weights))
# 61.28361802372382
Explanations of code:
First we build the list of states using Cartesian product itertools.product
For a given state i and die face f, we need to calculate j = state reached from i when adding f. I have two ways of calculating that, either as j = i + base[f] * (s[f] < targets[f]) or as j = lookup[s[:f] + (min(s[f]+1, targets[f]),) + s[f+1:]]. Because I'm paranoid, I calculated it both ways and checked that the two ways gave the same result. But you only need one way. You can remove lines j1 = ... to assert(j == j1) if you want.
Matrix P begins filled with zeroes, and we fill up to six cells per row with P[i, j] += p where p is probability of rolling face f.
Then we compute matrices Q and M as I indicated above.
We return the sum of all the cells on the first row of M.
To help you better understand what is going on, I encourage you to examine the values of all variables. For instance you could replace return M[0, :].sum() with return states, base, lookup, P, Q, I, M and then write states, base, lookup, P, Q, I, M = get_expected_n_trials(targets, dice_weights) in the python interactive shell, so that you can look at the variables individually.
A Monte-Carlo simulation:
Actually roll the die until we hit the requirements;
Count how many rolls we did;
Repeat experiment 1000 times to get the empirical average value.
Implementation in python:
from collections import Counter
from random import choices
from itertools import accumulate
from statistics import mean, stdev
dice_weights = [0.25, 0.05, 0.2, 0.15, 0.2, 0.15]
targets = [1, 3, 0, 1, 2, 3]
def avg_n_trials(targets, dice_weights, n_experiments=1000):
dice_faces = range(len(targets))
target_state = Counter(dict(enumerate(targets)))
cum_weights = list(accumulate(dice_weights))
results = []
for _ in range(n_experiments):
state = Counter()
while not state >= target_state:
f = choices(dice_faces, cum_weights=cum_weights)[0]
state[f] += 1
results.append(state.total()) # python<3.10: sum(state.values())
m = mean(results)
s = stdev(results, xbar=m)
return m, s
m, s = avg_n_trials(targets, dice_weights, n_experiments=10000)
print(m)
# 61.4044
Combinations without repetitions look like this, when the number of elements to choose from (n) is 5 and elements chosen (r) is 3:
0 1 2
0 1 3
0 1 4
0 2 3
0 2 4
0 3 4
1 2 3
1 2 4
1 3 4
2 3 4
As n and r grows the amount of combinations gets large pretty quickly. For (n,r) = (200,4) the number of combinations is 64684950.
It is easy to iterate the list with r nested for-loops, where the initial iterating value of each for loop is greater than the current iterating value of the for loop in which it is nested, as in this jsfiddle example:
https://dotnetfiddle.net/wHWK5o
What I would like is a function that calculates only one combination based on its index. Something like this:
tuple combination(i,n,r) {
return [combination with index i, when the number of elements to choose from is n and elements chosen is r]
Does anyone know if this is doable?
You would first need to impose some sort of ordering on the set of all combinations available for a given n and r, such that a linear index makes sense. I suggest we agree to keep our combinations in increasing order (or, at least, the indices of the individual elements), as in your example. How then can we go from a linear index to a combination?
Let us first build some intuition for the problem. Suppose we have n = 5 (e.g. the set {0, 1, 2, 3, 4}) and r = 3. How many unique combinations are there in this case? The answer is of course 5-choose-3, which evaluates to 10. Since we will sort our combinations in increasing order, consider for a minute how many combinations remain once we have exhausted all those starting with 0. This must be 4-choose-3, or 4 in total. In such a case, if we are looking for the combination at index 7 initially, this implies we must subtract 10 - 4 = 6 and search for the combination at index 1 in the set {1, 2, 3, 4}. This process continues until we find a new index that is smaller than this offset.
Once this process concludes, we know the first digit. Then we only need to determine the remaining r - 1 digits! The algorithm thus takes shape as follows (in Python, but this should not be too difficult to translate),
from math import factorial
def choose(n, k):
return factorial(n) // (factorial(k) * factorial(n - k))
def combination_at_idx(idx, elems, r):
if len(elems) == r:
# We are looking for r elements in a list of size r - thus, we need
# each element.
return elems
if len(elems) == 0 or len(elems) < r:
return []
combinations = choose(len(elems), r) # total number of combinations
remains = choose(len(elems) - 1, r) # combinations after selection
offset = combinations - remains
if idx >= offset: # combination does not start with first element
return combination_at_idx(idx - offset, elems[1:], r)
# We now know the first element of the combination, but *not* yet the next
# r - 1 elements. These need to be computed as well, again recursively.
return [elems[0]] + combination_at_idx(idx, elems[1:], r - 1)
Test-driving this with your initial input,
N = 5
R = 3
for idx in range(choose(N, R)):
print(idx, combination_at_idx(idx, list(range(N)), R))
I find,
0 [0, 1, 2]
1 [0, 1, 3]
2 [0, 1, 4]
3 [0, 2, 3]
4 [0, 2, 4]
5 [0, 3, 4]
6 [1, 2, 3]
7 [1, 2, 4]
8 [1, 3, 4]
9 [2, 3, 4]
Where the linear index is zero-based.
Start with the first element of the result. The value of that element depends on the number of combinations you can get with smaller elements. For each such smaller first element, the number of combinations with first element k is n − k − 1 choose r − 1, with potentially some of-by-one corrections. So you would sum over a bunch of binomial coefficients. Wolfram Alpha can help you compute such a sum, but the result still has a binomial coefficient in it. Solving for the largest k such that the sum doesn't exceed your given index i is a computation you can't do with something as simple as e.g. a square root. You need a loop to test possible values, e.g. like this:
def first_naive(i, n, r):
"""Find first element and index of first combination with that first element.
Returns a tuple of value and index.
Example: first_naive(8, 5, 3) returns (1, 6) because the combination with
index 8 is [1, 3, 4] so it starts with 1, and because the first combination
that starts with 1 is [1, 2, 3] which has index 6.
"""
s1 = 0
for k in range(n):
s2 = s1 + choose(n - k - 1, r - 1)
if i < s2:
return k, s1
s1 = s2
You can reduce the O(n) loop iterations to O(log n) steps using bisection, which is particularly relevant for large n. In that case I find it easier to think about numbering items from the end of your list. In the case of n = 5 and r = 3 you get choose(2, 2)=1 combinations starting with 2, choose(3,2)=3 combinations starting with 1 and choose(4,2)=6 combinations starting with 0. So in the general choose(n,r) binomial coefficient you increase the n with each step, and keep the r. Taking into account that sum(choose(k,r) for k in range(r,n+1)) can be simplified to choose(n+1,r+1), you can eventually come up with bisection conditions like the following:
def first_bisect(i, n, r):
nCr = choose(n, r)
k1 = r - 1
s1 = nCr
k2 = n
s2 = 0
while k2 - k1 > 1:
k3 = (k1 + k2) // 2
s3 = nCr - choose(k3, r)
if s3 <= i:
k2, s2 = k3, s3
else:
k1, s1 = k3, s3
return n - k2, s2
Once you know the first element to be k, you also know the index of the first combination with that same first element (also returned from my function above). You can use the difference between that first index and your actual index as input to a recursive call. The recursive call would be for r − 1 elements chosen from n − k − 1. And you'd add k + 1 to each element from the recursive call, since the top level returns values starting at 0 while the next element has to be greater than k in order to avoid duplication.
def combination(i, n, r):
"""Compute combination with a given index.
Equivalent to list(itertools.combinations(range(n), r))[i].
Each combination is represented as a tuple of ascending elements, and
combinations are ordered lexicograplically.
Args:
i: zero-based index of the combination
n: number of possible values, will be taken from range(n)
r: number of elements in result list
"""
if r == 0:
return []
k, ik = first_bisect(i, n, r)
return tuple([k] + [j + k + 1 for j in combination(i - ik, n - k - 1, r - 1)])
I've got a complete working example, including an implementation of choose, more detailed doc strings and tests for some basic assumptions.
I have the following data:
y-n-y-y-n-n-n
This repeats infinitely, such as:
y-n-y-y-n-n-n-y-n-y-y-n-n-n-y-n-y-y-n-n-n...
I have 5 "x".
"x" only sticks with "y".
Meaning, if I distribute x on the loop above, it will be:
y-n-y-y-n-n-n-y-n-y-y-n-n-n
x---x-x-----x-x
I want to count how many of the loop's element I needed to use to spread 5 x across, and the answer is 10.
How do I calculate it with a formula?
I presume what you're saying is that you need to process the first 10 elements of the infinite list to get 5 Y's, which match/stick with the 5 X's you have.
y-n-y-y-n-n-n-y-n-y-y-n-n-n-y-n-y-y-n-n-n...
x-_-x-x-_-_-_-x-_-x
^
L____ 10 elements read from the infinite list to place the 5 x's.
I also presume that your question is: given an input of 5 Xs, what is the number of elements you need to process in the infinite list to match those 5 Xs.
You could calculate it with a loop like the following pseudo-code:
iElementsMatchedCounter = 0
iXsMatchedCounter = 0
iXLimit = 5
strElement = ""
if (InfiniteList.IsEmpty() == false)
{
do
{
strElement = InfiniteList.ReadNextElement()
if (strElement == "y")
{
iXsMatchedCounter += 1
}
iElementsMatchedCounter += 1
} while ( (InfiniteList.IsEndReached() == false) AND (iXsMatchedCounter < iXLimit) )
}
if (iXsMatchedCounter = iXLimit)
then Print(iElementsMatchedCounter)
else Print("End of list reached before all X's were matched!")
The drawback of the above approach is that you are actually reading the infinite list, which might not be preferable.
Instead, given you know your list is an infinitely repeating sequence of the same elements y-n-y-y-n-n-n, you don't even need to loop through the entire list, but just operate on the sub-list y-n-y-y-n-n-n. The following algorithm describes how:
Given your starting input:
iNumberOfXs = 5 (you have 5 Xs to match)
iNumberOfYsInSubList = 3
(you have 3 Ys in the sub-list, the total list repeats infinitely)
iLengthOfSubList = 7 (you have 7 elements in the sub-list
y-n-y-y-n-n-n)
We then have intermediate results which are calculated:
iQuotient
iPartialLengthOfList
iPendingXs
iPendingLengthOfList
iResult
The following steps should give the result:
Divide the iNumberOfXs by iNumberOfYsInSubList. Here, this gives us 5/3 = 1.666....
Discard the remainder of the result (the 0.666...), so you're left with 1 as iQuotient. This is the number of complete sub-lists you have to iterate.
Multiply this quotient 1 with iLengthOfSubList, giving you 1*7=7 as iPartialLengthOfList. This is the partial sum of the result, and is the number of elements in the complete sub-lists you iterate.
Also multiply the quotient with iNumberOfYsInSubList, and subtract this product from iNumberOfXs, i.e. iNumberOfXs - (iQuotient * iNumberOfYsInSubList) = 5 - (1 * 3) = 2. Save this value 2 as iPendingXs, which is the number of as-yet unmatched X's.
Note that iPendingXs will always be less than iLengthOfSubList (i.e. it is a modulo, iPendingXs = iNumberOfXs MODULO iNumberOfYsInSubList).
Now you have the trivial problem of matching 2 X's (i.e. the value of iPendingXs calculated above) in the sub-list of y-n-y-y-n-n-n.
The pending items to match (counted as iPendingLengthOfList) is:
Equal to iPendingXs if iPendingXs is 0 or 1
Equal to iPendingXs + 1 otherwise (i.e. if iPendingXs is greater than 1)
In this case, iPendingLengthOfList = 3, because iPendingXs is greater than 1.
The sum of iPartialLengthOfList (7) and iPendingLengthOfList (3) is the answer, namely 10.
In general, if your sub-list y-n-y-y-n-n-n is not pre-defined, then you cannot hard-code the rule in step 6, but will instead have to loop through only the sub-list once to count the Ys and elements, similar to the pseudo-code given above.
When it comes to actual code, you can use integer division and modulo arithmetic to quickly to the operations in steps 2 and 4 respectively.
iQuotient = iNumberOfXs / iNumberOfYsInSubList // COMMENT: here integer division automatically drops the remainder
iPartialLengthOfList = iQuotient * iLengthOfSubList
iPendingXs = iNumberOfXs - (iQuotient * iNumberOfYsInSubList)
// COMMENT: can use modulo arithmetic like the following to calculate iPendingXs
// iPendingXs = iNumberOfXs % iNumberOfYsInSubList
// The following IF statement assumes the sub-list to be y-n-y-y-n-n-n
if (iPendingXs > 1)
then iPendingLengthOfList = iPendingXs + 1
else iPendingLengthOfList = iPendingXs
iResult = iPartialLengthOfList + iPendingLengthOfList
I'm writing a VBScript that sends out a weekly email with client activity. Here is some sample data:
a b c d e f g
2,780 2,667 2,785 1,031 646 2,340 2,410
Since this is email, I don't want a chart with a trend line. I just need a simple function that returns "up", "down" or "stable" (though I doubt it will ever be perfectly stable).
I'm terrible with math so I don't even know where to begin. I've looked at a few other questions for Python or Excel but there's just not enough similarity, or I don't have the knowledge, to apply it to VBS.
My goal would be something as simple as this:
a b c d e f g trend
2,780 2,667 2,785 1,031 646 2,340 2,410 ↘
If there is some delta or percentage or other measurement I could display that would be helpful. I would also probably want to ignore outliers. For instance, the 646 above. Some of our clients are not open on the weekend.
First of all, your data is listed as
a b c d e f g
2,780 2,667 2,785 1,031 646 2,340 2,410
To get a trend line you need to assign a numerical values to the variables a, b, c, ...
To assign numerical values to it, you need to have little bit more info how data are taken. Suppose you took data a on 1st January, you can assign it any value like 0 or 1. Then you took data b ten days later, then you can assign value 10 or 11 to it. Then you took data c thirty days later, then you can assign value 30 or 31 to it. The numerical values of a, b, c, ... must be proportional to the time interval of the data taken to get the more accurate value of the trend line.
If they are taken in regular interval (which is most likely your case), lets say every 7 days, then you can assign it in regular intervals a, b, c, ... ~ 1, 2, 3, ... Beginning point is entirely your choice choose something that makes it very easy. It does not matter on your final calculation.
Then you need to calculate the slope of the linear regression which you can find on this url from which you need to calculate the value of b with the following table.
On first column from row 2 to row 8, I have my values of a,b,c,... which I put 1,2,3, ...
On second column, I have my data.
On third column, I multiplied each cell in first column to corresponding cell in second column.
On fourth column, I squared the value of cell of first column.
On row 10, I added up the values of the above columns.
Finally use the values of row 10.
total_number_of_data*C[10] - A[10]*B[10]
b = -------------------------------------------
total_number_of_data*D[10]-square_of(A[10])
the sign of b determines what you are looking for. If it's positive, then it's up, if it's negative, then it's down, and if it's zero then stable.
This was a huge help! Here it is as a function in python
def trend_value(nums: list):
summed_nums = sum(nums)
multiplied_data = 0
summed_index = 0
squared_index = 0
for index, num in enumerate(nums):
index += 1
multiplied_data += index * num
summed_index += index
squared_index += index**2
numerator = (len(nums) * multiplied_data) - (summed_nums * summed_index)
denominator = (len(nums) * squared_index) - summed_index**2
if denominator != 0:
return numerator/denominator
else:
return 0
val = trend_value([2781, 2667, 2785, 1031, 646, 2340, 2410])
print(val) # -139.5
in python:
def get_trend(numbers):
rows = []
total_numbers = len(numbers)
currentValueNumber = 1
n = 0
while n < len(numbers):
rows.append({'row': currentValueNumber, 'number': numbers[n]})
currentValueNumber += 1
n += 1
sumLines = 0
sumNumbers = 0
sumMix = 0
squareOfs = 0
for k in rows:
sumLines += k['row']
sumNumbers += k['number']
sumMix += k['row']*k['number']
squareOfs += k['row'] ** 2
a = (total_numbers * sumMix) - (sumLines * sumNumbers)
b = (total_numbers * squareOfs) - (sumLines ** 2)
c = a/b
return c
trendValue = get_trend([2781,2667,2785,1031,646,2340,2410])
print(trendValue) # output: -139.5
Suppose there is number s=12 , now i want to make sequence with the element a1+a2+.....+an=12.
The criteria is as follows-
n must be minimum.
a1 and an must be 1;
ai can differs a(i-1) by only 1,0 and -1.
for s=12 the result is 6.
So how to find the minimum value of n.
Algorithm for finding n from given s:
1.Find q = FLOOR( SQRT(s-1) )
2.Find r = q^2 + q
3.If s <= r then n = 2q, else n = 2q + 1
Example: s = 12
q = FLOOR( SQRT(12-1) ) = FLOOR(SQRT(11) = 3
r = 3^2 + 3 = 12
12 <= 12, therefore n = 2*3 = 6
Example: s = 160
q = FLOOR( SQRT(160-1) ) = FLOOR(SQRT(159) = 12
r = 12^2 + 12 = 156
159 > 156, therefore n = 2*12 + 1 = 25
and the 25-numbers sequence for
159: 1,2,3,4,5,6,7,8,9,10,10,10,9,10,10,10,9,8,7,6,5,4,3,2,1
Here's a way to visualize the solution.
First, draw the smallest triangle (rows containing successful odd numbers of stars) that has a greater or equal number of stars to n. In this case, we draw a 16-star triangle.
*
***
*****
*******
Then we have to remove 16 - 12 = 4 more stars. We do this diagonally starting from the top.
1
**2
****3
******4
The result is:
**
****
******
Finally, add up the column heights to get the final answer:
1, 2, 3, 3, 2, 1.
There are two cases: s odd and s even. When s is odd, you have the sequence:
1, 2, 3, ..., (s-1)/2, (s-1)/2, (s-1)/2-1, (s-1)/2-2, ..., 1
when n is even you have:
1, 2, 3, ..., s/2, s/2-1, s/2-2, ..., 1
The maximum possible for any given series of length n is:
n is even => (n^2+2n)/4
n is odd => (n+1)^2/4
These two results are arrived at easily enough by looking at the simple arithmetic sum of series where in the case of n even it is twice the sum of the series 1...n/2. In the case of n odd it is twice the sum of the series 1...(n-1)/2 and add on n+1/2 (the middle element).
Clearly you can generate any positive number that is less than this max as long as n>3.
So the problem then becomes finding the smallest n with a max greater than your target.
Algorithmically I'd go for:
Find (sqrt(4*s)-1) and round up to the next odd number. Call this M. This is an easy to work out value and will represent the lowest odd n that will work.
Check M-1 to see if its max sum is greater than s. If so then that your n is M-1. Otherwise your n is M.
Thank all you answer me. I derived a simpler solution. The algorithm looks like-
First find what is the maximum sum that can be made using n element-
if n=1 -> 1 sum=1;
if n=2 -> 1,1 sum=2;
if n=3 -> 1,2,1 sum=4;
if n=4 -> 1,2,2,1 sum=6;
if n=5 -> 1,2,3,2,1 sum=9;
if n=6 -> 1,2,3,3,2,1 sum=12;
So from observation it is clear that form any number,n 9<n<=12 can be
made using 6 element, similarly number
6<n<=9 can be made at using 5 element.
So it require only a binary search to find the number of
element that make a particular number.