Question

I: The total number of ways in which a selection (one or more) can be made of $p+q+r$ things of which $p$ are alike, $q$ are alike, $r$ are alike is $(p+1)(q+1)(r+1)-1$
II: The number of permutations of n things taken together when $p$ of the things are alike of one kind, $q$ of them alike of a second kind, $r$ of them alike of a third kind and the rest all different is $\frac{n!}{p!q!r!3!}$
Which of the above statements is true?

• A. only I
• B. only II
• C. Both I and II
• D. Neither I nor II

Answer 1

The correct option is A only I

Permutations when all the objects are not distinct

The number of permutations of $n$ things taken all at a time when $p_1$ are of one kind, $p_2$ are of second kind and so on $p_n$ are of another kind and rest are all different $=\frac{n!}{p_1!p_2!\dots p_n!}$

Corollary: If $p$ things are alike of one kind, $q$ things are alike of a second kind, $r$ things are alike of third kind, the number of permutations $=\frac{(p+q+r)!}{p!q!r!}$.

Number of ways in which atleast one object may be selected out of $p$ alike objects of one type, $q$ alike objects of second type and $r$ alike of third type is $(p+1)(q+1)(r+1)–1$

Hence, only $\text{I}$ is true.