Question

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 ${ }^x C_y$ then -

• A. $x=m+n, y=m$
• B. $x=m+n+1, y=m$
• C. $x=m+n+1, y=m+1$
• D. $x=m+n, y=n$

Answer 1

Answer: (B)

Case 1 : When all $n$ red balls are taken but no green ball.
Only 1 arrangement is possible.
Case $2: n$ red balls and 1 green balls


${ }^{n+1} C_0+{ }^{n+1} C_1+{ }^{n+2} C_2+\ldots \ldots \ldots \ldots+{ }^{n+m} C_m$
${ }^{n+2} C_1+{ }^{n+2} C_2+\ldots \ldots \ldots+{ }^{n+m} C_m $
$\left(\therefore{ }^n C_r+{ }^n C_{r-1}={ }^{n+1} C_r\right)$
${ }^{n+3} C_2+{ }^{n+3} C_3+\ldots \ldots \ldots+{ }^{n+m} C_m$
Finally we get the sum as: ${ }^{ m + n +1} C _{ m }$