ProbabilityHardBloom L3
Question
Two numbers $x$ and $y$ are chosen at random without replacement from the set $\{ 1,2,3,4,\ldots\ldots.,100\}$. Then the probability that $x^{4} - y^{4}$ is divisible by 5 is
Options
A.$\frac{67}{99}$
B.$\frac{334}{495}$
C.$\frac{62}{99}$
D.$\frac{37}{99}$
Solution
Sol. [Key Concept: Two numbers $x,y$ without replacement: probability of the condition.]
$$x{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x}^{4}$$
$\begin{matrix} 5k & 5\lambda \\ 5k + 1 & 5\lambda + 1 \\ 5k + 2 & 5\lambda + 1 \\ 5k + 3 & 5\lambda + 1 \\ 5k + 4 & 5\lambda + 1 \end{matrix}$
$$\begin{matrix} P & \ = \frac{\ ^{80}C_{2} + \ ^{20}C_{2}}{\ ^{100}C_{2}} \\ & \ = \frac{20 \times 19 + 80 \times 79}{100 \times 99} = \frac{19 + 316}{5 \times 99} \\ & \ = \frac{335}{5 \times 99} = \frac{67}{99} \end{matrix}$$
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