Question

Let $f:A\to B$ be a function where set A contains 4 elements and set $B$ contains 3 elements. Number of functions defined from $A\to B$ which are not surjective is also equal to

• A. number of natural solution of the equation $x+y+z=11$.
• B. number of ways in which 10 children can be divided into two groups one containing 2 and other containing 8 children.
• C. number of ways in which 4 boys and 4 girls can be arranged alternately in a circle.
• D. total number of divisors of the number 3600 .

Answer 1

Answer: (A), (B), (D)

Total functions $=3^4=81$
Now, into functions $=$ Total functions - onto functions $=81-\frac{4!}{1!1!2!}\cdot \frac{1}{2!}\times 3!$
$=81-36=45$
(A)$x+y+z=11(x,y,z\in N)\Rightarrow x^{\prime }+y^{\prime }+z^{\prime }=8$
(B)
${}^{10}{C}_2=45\Rightarrow$(B) is correct.
(C) $3!\times 4!=144\Rightarrow$(C) is incorrect.
(D) $3600=2^4\cdot 3^2\cdot 5^2$
$\therefore$ Total divisors $=5\times 3\times 3=45\Rightarrow$(D) is correct.