Question

An unbiased die, with faces numbered 1, 2, 3, 4, 5, 6, is thrown n times and the list of n numbers shown up is noted. Then the probability that, among the numbers 1, 2, 3, 4, 5, 6, only three numbers appear in this list, is -

• A.
• B.
• C.
• D. None of these

Answer 1

Answer: (B)

Answer 2

Let us define a onto function F from

A : [r₁, r₂, …. , r_n] to B : [1, 2, 3] where r₁, r₂, …., r_n are the readings of the n throws and 1, 2 and 3 are the numbers that appear in the

n-throws. Number of such functions

M = N – [n(1) – n(2) + n(3)] where N = total number of functions and n (t) = number of functions having exactly t elements in the range. N = 3ⁿ, n (1) = 3. 2ⁿ, n(2) = 3, n(3) = 0 Ž M = (3ⁿ – 3 . 2ⁿ + 3) Hence total number of favourable cases = (3ⁿ – 3 . 2ⁿ + 3) . ⁶C₃

Total number of cases = 6ⁿ

\Required probability= $\frac { \left( 3 ^ { \mathrm { n } } - 3.2 ^ { \mathrm { n } } + 3 \right) \times { } ^ { 6 } \mathrm { C } _ { 3 } } { 6 ^ { \mathrm { n } } }$

Hence (2) is the correct answer