# Combinatorics Doubt

while calculating nCr (repetition is allowed) , I found that if r>n/2 then we can simply write r = n-r. Can someone please explain why or provide some links where i can find it. Thanks!

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nCr = nCn-r . Selecting r objects out of n is same as selecting n-r objects and throwing them away. You end up having r objects.

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ncr is number of ways of choosing r objects out of n
by choosing r elementsâ€¦u have automatically chosen n-r elements
by choosing who is in ur team â€¦u have automatically chosen who is not in ur team
so C(n,r)=C(n,n-r)

so if r>n/2 then n-r is smaller than r

so people calculate that

this is a bad proof but u can prove mathematically

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https://youtu.be/KbB0FjPg0mw watch last 15 minutes

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Write the expression for ^nC_{n-r} in expanded form and you will get ^nC_{r} as shown below:

^nC_{n-r} = \frac{n!} {(n-r)! \times (n - (n-r))!} = \frac{n!} {(n - r)! \times r!} = ^nC_{r}

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First of all, for nCr, it doesnâ€™t matters that elements are distinct or repeated.
This is only The number of ways to choose â€śrâ€ť elements out of â€śnâ€ť elements.
for example consider an sequence X = {a,b,c,d}, here n = 4 and u are asked that how many ways u can select 2 elements from this array, then your answer will be = 4C2 = 6.
these are the possible ways : {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}
Note that above pairs do not describe about properties of elements, instead these are indexes (say first and second, first and thirdâ€¦like this).
The mathematical expression of nCr = n! / (r! *(n-r)!) and nC(n-r) = n! / (r!* (n-r)!).
Both are same.

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