Let be a set with exactly 5 elements and be a set with exactly 7 elements. If is the number of one-one functions from to and is the number of onto functions from to , then the value of is ________. [2018]

Question

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 $X$ to $Y$ and $β$ is the number of onto functions from $Y$ to $X$ , then the value of $\frac{1}{5!} (β - α)$ is ________. [2018]

Smart Achievers · Accepted Answer

(119)

$Here n (X) = 5 and n (Y) = 7$

$Number of one-one functions = α = {}^{7}C_{5} \times 5!$

$and$ $Number of onto functions Y \to X = β$

$= \frac{7!}{3! 4!} \times 5! + \frac{7!}{{(2!)}^{3} 3!} \times 5! = ({}^{7}C_{3} + 3 \times {}^{7}C_{3}) 5!$

$= 4 \times {}^{7}C_{3} \times 5!$

$\Rightarrow \frac{β - α}{5!} = 4 \times {}^{7}C_{3} - {}^{7}C_{5} = 4 \times 35 - 21 = 119$

Solution

Q.
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 $X$ to $Y$ and $β$ is the number of onto functions from $Y$ to $X$ , then the value of $\frac{1}{5!} (β - α)$ is ________. [2018]

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Solution

Q. 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 to Y and β is the number of onto functions from Y to X, then the value of 15!(β-α) is ________. [2018]

Ans. (119) Here n(X)=5 and n(Y)=7 Number of one-one functions=α=C57×5! and Number of onto functions Y→X=β =7!3! 4!×5!+7!(2!)3 3!×5!=(C37+3×C37)5! =4×C37×5! ⇒ β-α5!=4×C37-C57=4×35-21=119

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Q.
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 $X$ to $Y$ and $β$ is the number of onto functions from $Y$ to $X$ , then the value of $\frac{1}{5!} (β - α)$ is ________. [2018]