Official PYQ · 1993Q8.6Sets, Relations and FunctionsMCQ1 mark

GATE CS 1993 Q8.6 — Sets, Relations and Functions

Let $A$ and $B$ be sets with cardinalities $m$ and $n$ respectively. The number of one-one mappings from $A$ to $B$ , when $m < n$ , is

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Answer: B

To find the number of one-one (injective) mappings from set $A$ to set $B$ , where $|A| = m$ and $|B| = n$ with $m < n$ :

For a mapping to be one-one, each element in the domain $A$ must map to a unique and distinct element in the codomain $B$ .

The first element of $A$ can be mapped to any of the $n$ elements in $B$ ( $n$ choices).
The second element of $A$ must be mapped to a different element, leaving $n - 1$ choices.
Continuing this process, the $m$ -th element of $A$ will have $n - (m - 1) = n - m + 1$ choices.

By the product rule, the total number of one-one mappings is:

\text{Total mappings} = n \times (n - 1) \times (n - 2) \times \dots \times (n - m + 1)

This expression represents the number of permutations of $n$ items taken $m$ at a time, which is denoted as:

^nP_m = \frac{n!}{(n - m)!}

Hence, the correct option is B.

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Published 10 Oct 2026

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