Question

Let x denote the total number of one-one functions from a set A with 3 elements to a set B with 5 elements and y denote the total number of one-one functions from the set A to the set A $\times$ B. Then :

• A. 2y = 91x
• B. 2y = 273x
• C. y = 91x
• D. y = 273x

Hint:

The number of one-to-one functions from a set of size 'm' to a set of size 'n' is $^n P_m$. This can be thought of as choosing 'm' distinct images from 'n' possibilities and arranging them, which is the definition of permutation.

Answer 1

The Correct Option is A

\[ x = {}^{5}P_3 = 5\cdot4\cdot3 = 60 \] \[ |A\times B| = 3\times5 = 15 \] \[ y = {}^{15}P_3 = 15\cdot14\cdot13 = 2730 \] \[ \frac{y}{x}=\frac{2730}{60}=\frac{91}{2} \Rightarrow 2y=91x \] \[ \boxed{2y=91x} \]