Ari chooses 7 balls at random from n balls numbered 1 to n . If the probability that no two of the drawn balls have consecutive numbers equals the probability of exactly one pair of consecutive numbers in the chosen balls, find n .

Step-by-step solution

Case I. When no two selected balls have consecutive numbers. aligned & x₁+x₂+x₃+ .+x₈=n-7 & x₁+x₂+x₃ .+x₇+x₈=n-13 aligned Total ways = ^ n-6 C₇ Case II. When exactly two balls have consecutive numbers x₁+x₂+x₃+ . x₈=n-7 Any one out of x₂, x₃, , x₇ must be zero aligned & x₁+ X₂+X₃+ +X₇ _ any 5 x_i ^ s +x₈=n-12 & Total ways = ^6 C₁ n- ^6 C₆ & ^ n-6 C₇= ^6 C₁ ^ n-6 C₆ & n-12=7 6 & n=54 aligned