In an examination, there are 10 true-false type questions. Out of 10 , a student can guess the answer of 4 questions correctly with probability 3 4 and the remaining 6 questions correctly with probability 1 4 . If the probability that the student guesses the answers of exactly 8 questions correctly out of 10 is 27 k 4 10 , then k is equal to

Step-by-step solution

Given, A = 1 , 2 , 3 , 4 : P A = 3 4 → Correct answer B = 5 , 6 , 7 , 8 , 9 , 10 ; P B = 1 4 Correct answer Now total ways in which 8 Correct answer is given will be, ⇒ Case (i)   4 , 4 : C 4 4 3 4 4 · C 4 6 · 1 4 4 · 3 4 2 ⇒ Case   (ii)  3 , 5 : C 3 4 3 4 3 · 1 4 1 · C 5 6 1 4 5 · 3 4 ⇒ Case (iii)   2 , 6 : C 2 4 3 4 2 1 4 2 · C 6 6 1 4 6 Total probability = 1 4 10 3 4 × 15 × 3 2 + 4 × 3 3 × 6 × 3