Let N denotes the sum of the numbers obtained when two dice are rolled. If the probability that $2^{\mathrm{N}}<\mathrm{N}!$ is $\frac{m}{n}$, where m and n are coprime, then $4 m-3 n$ is equal to

JEE Main · 2023 · 10 Apr 2023 · Shift 1 · Probability

Let N denotes the sum of the numbers obtained when two dice are rolled. If the probability that

2^{N} < N!

\frac{m}{n}

, where m and n are coprime, then

4 m - 3 n

is equal to

Answer: (A)

When two dice are thrown the sum

N

runs from

2

12

. Testing

2^{N} < N!

: for

N = 2

4 \neq < 2

; for

N = 3

8 \neq < 6

; and for every

N \geq 4

the inequality holds. So it fails only when

N = 2

N = 3

. Now

P (N = 2) = \frac{1}{36}

and

P (N = 3) = \frac{2}{36}

, so

P (N \leq 3) = \frac{3}{36} = \frac{1}{12}

and

P (2^{N} < N!) = 1 - \frac{1}{12} = \frac{11}{12}

. Thus

m = 11, n = 12

, giving

4 m - 3 n = 44 - 36 = 8

JEE Main · 2023 · 10 Apr 2023 · Shift 1 · Single correct option