JEE Main 2026 — Apr 4, Shift 1 — Mathematics solution

JEE Main2026Apr 4, Shift 1Mathematics

A coin is tossed $8$ times. If the probability that exactly $4$ heads appear in the first six tosses and exactly $3$ heads appear in the last five tosses is $p$ , then $96 p$ is equal to _______.

Your answer

Solution

Label tosses $x_{1}, x_{2}, \dots, x_{8}$ . The event requires $4$ heads among ${x_{1}, \dots, x_{6}}$ and $3$ heads among ${x_{4}, \dots, x_{8}}$ . Let $k$ denote the number of heads in the overlap ${x_{4}, x_{5}, x_{6}}$ .

For each $k$ , count sequences as $(k 3)$ for the overlap times $(4 - k 3)$ for ${x_{1}, x_{2}, x_{3}}$ times $(3 - k 2)$ for ${x_{7}, x_{8}}$ . Valid values are $k = 1, 2, 3$ :

$k = 1$ : $(1 3) (3 3) (2 2) = 3 \cdot 1 \cdot 1 = 3.$

$k = 2$ : $(2 3) (2 3) (1 2) = 3 \cdot 3 \cdot 2 = 18.$

$k = 3$ : $(3 3) (1 3) (0 2) = 1 \cdot 3 \cdot 1 = 3.$

Total favourable: $24$ . So $p = \frac{24}{2 ^{8}} = \frac{24}{256}$ , and

$96 p = \frac{96 \cdot 24}{256} = 9.$

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