Let $X$ be a set with exactly 5 elements and $Y$ be a set with exactly 7 elements. If $\alpha$ is the number of one-one functions from $X$ to $Y$ and $\beta$ is the number of onto functions from $Y$ to $X$, then the value of $\frac{1}{5!}(\beta-\alpha)$ is ____.

JEE Advanced · 2018 · 20 May 2018 · Paper 2 · Permutations and Combinations

Let

X

be a set with exactly 5 elements and

Y

be a set with exactly 7 elements. If

α

is the number of one-one functions from

X

Y

and

β

is the number of onto functions from

Y

X

, then the value of

\frac{1}{5 !} (β - α)

is ____.

Answer: 119

Solution

Number of one-one functions from

X

Y

α =^{7} P_{5} =^{7} C_{5} \cdot 5! = 21 \cdot 120 = 2520

Number of onto functions from

Y

X

using inclusion-exclusion:

β = 5^{7} -^{5} C_{1} \cdot 4^{7} +^{5} C_{2} \cdot 3^{7} -^{5} C_{3} \cdot 2^{7} +^{5} C_{4} \cdot 1^{7}

= 78125 - 5 (16384) + 10 (2187) - 10 (128) + 5 (1)

= 78125 - 81920 + 21870 - 1280 + 5 = 16800

\frac{β - α}{5 !} = \frac{16800 - 2520}{120} = \frac{14280}{120} = 119

JEE Advanced · 2018 · 20 May 2018 · Paper 2 · Numerical answer

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