Question

Let $A = \{a, b, c, d, e\}$ and $B = \{1, 2, 3\}$. Then the number of onto functions from set A to set B is

• A. 150
• B. 180
• C. 210
• D. 250

Answer 1

Answer: (A)

150

Answer 2

To find the number of onto functions from set A to set B, we use the formula for the number of surjective functions.

Let $n = |A|$ be the number of elements in set A, and $m = |B|$ be the number of elements in set B. In this question, $A = \{a, b, c, d, e\}$, so $n = 5$. And $B = \{1, 2, 3\}$, so $m = 3$.

The number of onto functions from a set of $n$ elements to a set of $m$ elements is given by the Principle of Inclusion-Exclusion:

$\sum_{k=0}^{m} (-1)^k \binom{m}{k} (m-k)^n$

Substitute $n=5$ and $m=3$ into the formula: Number of onto functions = $\binom{3}{0}(3-0)^5 - \binom{3}{1}(3-1)^5 + \binom{3}{2}(3-2)^5 - \binom{3}{3}(3-3)^5$ $= \binom{3}{0}3^5 - \binom{3}{1}2^5 + \binom{3}{2}1^5 - \binom{3}{3}0^5$

Now, calculate each term:

1. $\binom{3}{0}3^5 = 1 \times 243 = 243$
2. $\binom{3}{1}2^5 = 3 \times 32 = 96$
3. $\binom{3}{2}1^5 = 3 \times 1 = 3$
4. $\binom{3}{3}0^5 = 1 \times 0 = 0$ (since $5 \ge 1$)

Substitute these values back into the expression: Number of onto functions = $243 - 96 + 3 - 0$ $= 147 + 3$ $= 150$

Thus, there are 150 onto functions from set A to set B.