Question ID: #420
Number of functions $f:\{1,2,…,100\}\rightarrow\{0,1\}$, that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to:
Solution:
We need to define a function $f$ from Domain $D = \{1, 2, \dots, 100\}$ to Codomain $C = \{0, 1\}$.
The condition is that exactly one element from the subset $S = \{1, 2, \dots, 98\}$ is mapped to 1.
Assign values for elements in $S$
We need to choose exactly 1 element out of 98 to take the value 1. The remaining 97 elements must take the value 0.
Number of ways = ${}^{98}C_{1} = 98$.
Assign values for remaining elements
The remaining elements in the domain are $\{99, 100\}$. There are no restrictions on these.
$f(99)$ can be 0 or 1 (2 choices).
$f(100)$ can be 0 or 1 (2 choices).
Number of ways = $2 \times 2 = 4$.
Total functions
Total number of functions = (Ways for $S$) $\times$ (Ways for rest)
$$ = 98 \times 4 = 392 $$
Ans. (392)
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