There are n identical red balls & m identical green balls. The number of different linear arrangements consisting of "n red balls but not necessarily all the green balls" is `"^xC_y` then (A) x=m+n , y=m (B) x=m+n+1, y=m (C) x=m+n+1, y=m+1 (D) x=m+n, y=n

Options

A. `x=m+n`
B. `y=m`
C. `x=m+n+1`
D. `y=n`

Explanation

To solve the problem of finding the number of different linear arrangements consisting of "n red balls but not necessarily all the green balls," we can follow these steps:

### Step 1: Understand the Problem
We have n identical red balls and m identical green balls. We want to arrange all n red balls and some (or none) of the m green balls in a linear fashion.

### Step 2: Determine the Total Arrangements
When arranging n red balls and k green balls (where k can vary from 0 to m), the total number of balls in any arrangement will be \( n + k \). The number of ways to choose k green balls from m is given by the binomial coefficient \( \binom{m}{k} \).

...