Question

Let $A$ and $B$ be finite sets and $P_A$ and $P_B$ respectively denote their power sets. If $P_B$ has $112$ elements more than those in $P_{A^{\prime }}$ then the number of functions from $A$ to $B$ which are injective is

• A. 224
• B. 56
• C. 120
• D. 840

Answer 1

Answer: (D)

Let $n(A)=m$ and $n(B)=n$. Then, according to the question,
$n(P(B))-n(P(A))=112$
$\Rightarrow 2^n-2^m=112$
$\Rightarrow 2^m\left(2^{n-m}-1\right)=16\times 7$
$\Rightarrow 2^m\left(2^{n-m}-1\right)=2^4\left(2^3-1\right)$
$\therefore m=4$ and $n-m=3$
$\Rightarrow m=4,n=7$
$\therefore$ Number of injective functions from $A$ to $B$
$={}^{n(B)}{P}_{n(A)}={}^7{P}_4=\frac{7!}{3!}=840$