A = { x_1, x_2, x_3, x_4 }; , B = { y_1, y_2, | vedclass

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(B) The problem asks for the number of one-one functions $f: A 	o B$ such that $f(x_i) 
eq y_i$ for all $i \in \{1, 2, 3, 4\}$.Since the number of e

Vedclass · Accepted Answer

$9$

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(B) The problem asks for the number of one-one functions $f: A 	o B$ such that $f(x_i) 
eq y_i$ for all $i \in \{1, 2, 3, 4\}$.Since the number of elements in both sets is equal $(n=4)$,a one-one function is equivalent to a permutation of the set $\{y_1, y_2, y_3, y_4\}$.The condition $f(x_i) 
eq y_i$ means that no element is mapped to itself,which is the definition of a derangement.The number of derangements of $n$ objects is given by the formula $D_n = n! \left( 1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \dots + \frac{(-1)^n}{n!} ight)$.For $n = 4$:$D_4 = 4! \left( 1 - 1 + \frac{1}{2} - \frac{1}{6} + \frac{1}{24} ight)$$D_4 = 24 \left( \frac{1}{2} - \frac{1}{6} + \frac{1}{24} ight)$$D_4 = 24 \left( \frac{12 - 4 + 1}{24} ight) = 9$.Thus,there are $9$ such functions.

$A = \{ x_1, x_2, x_3, x_4 \}; \, B = \{ y_1, y_2, y_3, y_4 \}.$ $A$ function is defined from set $A$ to set $B.$ The number of one-one functions such that $f(x_i) \neq y_i$ for $i = 1, 2, 3, 4$ is equal to:

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