NIMCET 2014 — Mathematics PYQ

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Question

NIMCET 2014 — Mathematics PYQ

NIMCET | Mathematics | 2014

Suppose $r$ integers, $0 < r < 10$ , are chosen from $\{0, 1, 2, \dots, 9\}$ at random and with replacement. The probability that no two are equal, is:

Choose the correct answer:

Explanation

Calculation:

Given: $r$ integers, $0 < r < 10$ , are chosen from $\{0, 1, 2, \dots, 9\}$ at random and with replacement.

Total possible outcomes:

\text{Total Outcomes} = 10^{r}

Number of ways to choose $r$ integers such that no two integers are equal:

\text{Favorable Outcomes} = ^{10}C_{r} \times r! = ^{10}P_{r}

The probability that no two are equal is given by:

P = \frac{^{10}C_{r}}{10^{r}}

Substituting the formula for $^{10}C_{r}$ :

P = \frac{10!}{10^{r} \times r! \times (10 - r)!}