JEE Main · 2022 · 26 Jun 2022 · Shift 2 · Probability

If the probability that a randomly chosen 6-digit number formed by using digits 1 and 8 only is a multiple of 21 is

p

, then

96 p

is equal to ____.

Answer: 33

Solution

Total 6-digit numbers using only digits 1 and 8:

2^{6} = 64

.

A number is divisible by 21 if it is divisible by both 3 and 7. For divisibility by 3, the sum of digits must be divisible by 3.

Case I: All six digits are 1. Sum = 6, divisible by 3. Count = 1.

Case II: All six digits are 8. Sum = 48, divisible by 3. Count = 1.

Case III: Three 1's and three 8's. Sum =

3 (1) + 3 (8) = 27

, divisible by 3. Count =

(3 6) = 20

.

Total favorable outcomes =

1 + 1 + 20 = 22

.

Probability

p = \frac{22}{64} = \frac{11}{32}

.

Therefore,

96 p = 96 \times \frac{11}{32} = 33

JEE Main · 2022 · 26 Jun 2022 · Shift 2 · Integer answer

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