Question

If a set \(A\) contains \(3\) elements and another set \(B\) contains \(4\) elements, then the number of functions from \(A\) to \(B\), which are not injective, is

• A. \(24\)
• B. \(64\)
• C. \(40\)
• D. \(12\)

Hint:

For questions involving non-injective functions: \[ \text{Non-injective} = \text{Total Functions} - \text{Injective Functions} \] \[ =n^m-{}^{n}P_m \]

Answer 1

The Correct Option is C

Concept:

• Total number of functions from a set with \(m\) elements to a set with \(n\) elements is \(n^m\).

• Number of injective functions from \(A\) to \(B\) is \[ {}^{n}P_{m} \] provided \(n\ge m\).

Step 1: Find the total number of functions. \[ |A|=3,\qquad |B|=4 \] Hence, \[\begin{aligned} \text{Total functions} = 4^3 = 64 \end{aligned}\]

Step 2: Find the number of injective functions. \[\begin{aligned} {}^{4}P_{3} = \frac{4!}{(4-3)!} = 24 \end{aligned}\]

Step 3: Find the number of non-injective functions. \[\begin{aligned} 64-24 = 40 \end{aligned}\] \[\begin{aligned} \boxed{40} \end{aligned}\] Hence, option \(\mathbf{(C)}\) is correct.