Question

If a set \(A\) has \(m\)-elements and the set \(B\) has \(n\)-elements, then the number of injections from \(A\) to \(B\) is

• A. \({}^{n}C_{m}\), if \(n\geq m\)
• B. \({}^{n}P_{m}\), if \(n\geq m\)
• C. \(0\), if \(n\geq m\)
• D. \(m\cdot {}^{n}C_{m}\), if \(n\geq m\)

Hint:

For injections from a set with \(m\) elements to a set with \(n\) elements: \[ \text{Number of injections}={}^{n}P_{m} \] because distinct images are required for every element.

Answer 1

The Correct Option is B

Step 1: Recall the definition of injection.
An injection (one-one function) from set \(A\) to set \(B\) means that different elements of \(A\) must map to different elements of \(B\).
Thus, no two elements of \(A\) can have the same image in \(B\).

Step 2: Count the choices for each element of \(A\).
Suppose, \[ |A|=m \quad \text{and} \quad |B|=n \] For the first element of \(A\), there are \[ n \] choices in \(B\).
For the second element, since repetition is not allowed, \[ n-1 \] choices remain.
Similarly, continuing this process, the total number of injections becomes \[ n(n-1)(n-2)\cdots(n-m+1) \]

Step 3: Express in permutation form.
The above product is exactly \[ {}^{n}P_{m} = \frac{n!}{(n-m)!} \] provided \[ n\geq m \] If \[ n\lt m, \] then injection is not possible.

Step 4: Final conclusion.
Therefore, the number of injections from \(A\) to \(B\) is \[ \boxed{{}^{n}P_{m}} \] for \[ n\geq m \] Hence, the correct option is (2).