The number of functions f: {1, 2, ldots, 100} | vedclass

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(A) We are looking for the number of functions $f: \{1, 2, \ldots, 100\} \rightarrow \{0, 1\}$ such that exactly one element from the set $\{1, 2, \ld

Vedclass · Accepted Answer

$392$

Vedclass · Answer

(A) We are looking for the number of functions $f: \{1, 2, \ldots, 100\} ightarrow \{0, 1\}$ such that exactly one element from the set $\{1, 2, \ldots, 98\}$ is mapped to $1$.$1$. First,we choose exactly one element from the set $\{1, 2, \ldots, 98\}$ to be mapped to $1$. There are $\binom{98}{1} = 98$ ways to do this.$2$. The remaining $97$ elements in the set $\{1, 2, \ldots, 98\}$ must be mapped to $0$. There is only $1$ way to do this.$3$. The element $99$ can be mapped to either $0$ or $1$. There are $2$ choices for this.$4$. The element $100$ can be mapped to either $0$ or $1$. There are $2$ choices for this.By the multiplication principle,the total number of such functions is $98 	imes 1 	imes 2 	imes 2 = 392$.

The number of functions $f: \{1, 2, \ldots, 100\} \rightarrow \{0, 1\}$ that assign $1$ to exactly one of the positive integers less than or equal to $98$ is equal to $\qquad$.

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