Question

A card is drawn from a pack, the card is replaced & the pack shuffled. If this is done 6 times, the probability that the cards drawn are 2 hearts, 2 diamonds & 2 black cards is `p/q` then total number of proper divisors of (pq). Where p & q are co-prime natural numbers). Select the correct answer from above options

Answer 1

As we have to select `6` cards, Number of ways of selecting `2` hearts from deck of cards ` = C(6,2)*13*13` Now, number of ways of selecting `2` diamonds from deck of cards ` = C(4,2)*13*13` Now, number of ways of selecting `2` blacks from deck of cards ` = C(2,2)*26*126` `:.` Total probabilty of selecting these `6` cards ` = ((C(6,2)*13*13)(C(4,2)*13*13)(C(2,2)*26*26))/(52)^6 = 45/512` We are given this probability is equal to `p/q`. `:. p/q = 45/512` `=>pq = 45**512 = (3^2)(5^1)(2^9)` `:.` Number of proper divisors of `pq = (2+1)(1+1)(9+1)-2 = 58` Here, we have subtracted `2` to exclude `1` and `pq`.