Ari chooses 7 balls at random from n balls numbered 1 to n . If the probability that no two of the drawn balls have consecutive numbers equals the probability of exactly one pair of consecutive numbers in the chosen

Question

Quantrex Academy · Accepted Answer

Correct answer: 54

Ari chooses 7 balls at random from n balls numbered 1 to n . If the probability that no two of the drawn balls have

Case I. When no two selected balls have consecutive numbers.

\begin{aligned} & x_1+x_2+x_3+\ldots .+x_8=n-7 \\ & \Rightarrow x_1+x_2+x_3 \ldots .+x_7+x_8=n-13 \end{aligned}

Total ways

={ }^{n-6} C_7

Case II. When exactly two balls have consecutive numbers

x_1+x_2+x_3+\ldots . x_8=n-7

Any one out of

x_2, x_3, \ldots, x_7

must be zero

\begin{aligned} & x_1+\underbrace{X_2+X_3+\ldots \ldots+X_7}_{\text {any } 5 x_i{ }^{\prime} s}+x_8=n-12 \\ & \text { Total ways }={ }^6 C_1 \cdot n-{ }^6 C_6 \\ & \because \quad{ }^{n-6} C_7={ }^6 C_1 \cdot{ }^{n-6} C_6 \\ & \Rightarrow n-12=7 \times 6 \\ & \Rightarrow n=54 \end{aligned}

Ari chooses 7 balls at random from n balls numbered 1 to n . If the...

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