Question

Number of ways in which an ordered set of r whole numbers $(r\ge 2)$ can be selected that their sum is n is equal to

• A. ${}^{n+r-1}{C}_n$
• B. ${}^{n+1}{C}_{r+1}$
• C. ${}^{n+r-1}{C}_r$
• D. None of these

Answer 1

Answer: (A)

${}^{n+r-1}{C}_n$

Consider the sum n as $n$ identical objects
Now consider that all the $n$ objects are placed on a row.
Let us assume some separators that separates the n objects.

Now as we want to separate them into r group[as blank groups are also allowed],take the separators as objects.
Now we need $r-1$ separators to make r groups.Therefore total number of objects is $n+r-1$.

There will be $(r-1)$ places for the separators to occupy.

Therefore we can arrange the separators in ${}^{n+r-1}{C}_{r-1}$ways