Question

The combination coefficient ^{${}^{n-1}{C}_m$} denotes -

(a) The number of ways in which n things of which m are alike and rest different can arrange in a circle.

(b) The number of ways in which m different things can be selected out of n different thing if a particular thing is always excluded.

(c) Number of ways in which n different balls can be distributed in m different boxes so that no box remains empty and each box can hold any number of balls.

• A. onle option a
• B. only option b
• C. Both options a and b

• D. All a, b and c

Answer 1

The correct option is B

only option b

(a) The number of ways in which n things of which p are alike and rest different can arranged in a circle = $\frac{\left(\right(n-1)!)}{m!}$

(b) The number of ways in which m different things can be selected out of n different thing if a particular thing is always excluded = select m different things from n-1 things = ^{${}^{n-1}{C}_m$}

(c) Number of ways in which n different balls can be distributed in m different boxes each box can hold any number of balls = $m^n$

Explanation: 1^st ball can be put in any of the m boxes in m ways.

Similarly, $2^{nd}$ball, $3^{rd}$ ball,.... can be a put in any of the m boxes in m ways.
So, total number of ways = m x m x m x m.....(n times) = $m^n$