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).

Explanation

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)`

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