A man is dealt a poker hand (consisting of 5 cards) from an ordinary pack of 52 playing cards.
Find the number of ways in which he can be dealt a "straight" (a straight is five consecutive values not of the same suit, eg. {Ace, 2, 3, 4, 5}, {2, 3, 4, 5, 6}.......................... & {10, J, Q, K, Ace})
My approach is as follow There are five space in the first space the number of ways of selecting a card is 4 viz. ace, diamond , heart and spade Hence $4^5$ ways. Number of cases are 10 viz. {Ace, 2, 3, 4, 5}, {2, 3, 4, 5, 6}.......................... & {10, J, Q, K, Ace})
Hence the answer is $10*4^5$ but the actual answer is $10 (4^5-4)$. Please tell me my mistake.
A straight does not have cards of the same suit. Therefore you need to subtract those values which is $4 x 10$. So final answer is $$10 × 4 ^5-10×4$$