Question

In a packet there are m different books, n different pens and p different pencils. The number of selections of at least one article of each type from the packet is
(a) 2^{m + n + p} − 1
(b) (m + 1) (n + 1) (p + 1) − 1
(c) 2^{m + n + p}
(d) none of these.

Answer 1

(a) 2^{m + n + p} − 1
Each of the object, i.e book, pen or pencil, can either be selected or not selected.
So, each of the object will have two outcomes, i.e selection or rejection.
Number of ways of selecting all the books = 2$\times$2$\times$2$\times$.....m times = 2^m
Number of ways of selecting all the pens = 2ⁿ
Number of ways of selecting all the pencils = 2^p
Thus, by fundamental principle of counting, total number of ways = 2^m$\times$2ⁿ$\times$2^p = 2^m+n+p
But, of all these ways, there is one way in which all the objects are rejected, which is not valid. Hence, it needs to be subtracted.
∴ Total number of valid selections = 2^{m + n + p} − 1