JEE Advanced 2018 · Paper 2 · Mathematics Q9

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MathematicsNumerical+3Question 45 of 54

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 _____ .

Your answer

Answer and worked solution

Official answer: 119

One-one functions $X\to Y$ : $\alpha={}^7P_5=2520$ .

Onto functions $Y\to X$ , by inclusion-exclusion:
$\beta=\sum_{k=0}^5(-1)^k\binom5k(5-k)^7=78125-81920+21870-1280+5=16800.$
(This is $S(7,5)\cdot5!=140\times120$ .)

Value.
$\frac{\beta-\alpha}{5!}=\frac{16800-2520}{120}=119.$

Answer: 119

Topics:Types of functions Distribution and arrangements

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JEE Advanced 2018 · Paper 2, Mathematics question 9