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प्रश्न
Let X denote the number of times heads occur in n tosses of a fair coin. If P (X = 4), P (X= 5) and P (X = 6) are in AP, the value of n is
विकल्प
7, 14
10, 14
12, 7
14, 12
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उत्तर
7, 14
\[\text{ Here } , p = \frac{1}{2}\text{ and } q = \frac{1}{2}\]
\[\text{ Binomial distribution is given by } \]
\[P(X = r) =^{n}{}{C}_r \left( \frac{1}{2} \right)^r \left( \frac{1}{2} \right)^{n - r}\]
P (X = 4), P (X = 5), P(X = 6) are in A.P.
\[\therefore ^{n}{}{C}_4 +^{n}{}{C}_6 = 2 ^{n}{}{C}_5 \]
\[ \Rightarrow \frac{n(n - 1)(n - 2)(n - 3)}{2\left( 4! \right)} + \frac{n(n - 1)(n - 2)(n - 3)(n - 4)(n - 5)}{2\left( 6! \right)} = \frac{n(n - 1)(n - 2)(n - 3)(n - 4)}{5!}\]
\[\text{ By simplifying, we get} \]
\[\frac{1}{2} + \frac{(n - 4)(n - 5)}{2(30)} = \frac{n - 4}{5}\]
\[\text{ Taking LCM as 60, we get}\]
\[30 + n^2 - 9n + 20 = 12n - 48 \]
\[ \Rightarrow n^2 - 21n + 98 = 0\]
\[ \Rightarrow (n - 7)(n - 14) = 0\]
\[ \Rightarrow n = 7, 14\]
