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 Question from Aastha, a student: FIND THE VALUE OF:- the sum form r = 0 to r = m of C(n+r,n)=?

Aastha,

Think of a line of n+m+1 people. You are going to walk down the line and select n+1 of them. Where could the last person chosen be? If this last person is in the n+1st place then you could choose your n+1 people in C(n,n) ways (you must have chosen n of the first n people);

if this last person is in the n+2nd place then you could choose your n+1 people in C(n+1,n) ways (you must have chosen n of the first n+1 people);

if this last person is in the n+3rd place then you could choose your n+1 people in C(n+2,n) ways (you must have chosen n of the first n+2 people); …;

if this last person is in the n+m+1st place then you could choose your n+1 people in C(n+m,n) ways (you must have
chosen n of the first n+m people).

Thus your sum counts the number of ways to choose n+1 of n+m+1 people in a line which is … ?

Penny

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