Question

Let $f : A \rightarrow B$ be a function where set A contains 4 elements and set $B$ contains 3 elements. Number of functions defined from $A \rightarrow B$ 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.