Number of ways of selecting 5 cards from a deck of 52 cards such that one card of each suit is taken

To find the number of ways of selecting 5 cards from a standard deck of 52 cards such that one card of each suit is taken, we can break down the problem into separate cases for each suit (spades, hearts, diamonds, and clubs).

Case 1: Selecting 1 card from each suit
In this case, we need to select one card from each suit. Since there are 13 cards in each suit (Ace to King), the number of ways to select one card from each suit is:

13 * 13 * 13 * 13 = 28,561

Case 2: Selecting 4 cards from one suit and 1 card from another suit
In this case, we select 4 cards from one suit and 1 card from another suit. There are 4 suits to choose from and we can select any 2 suits for this case. For each suit, there are 13 cards to choose from. The number of ways to select 4 cards from one suit and 1 card from another suit is:

4C2 * 13C4 * 13C1 = 6 * 715 * 13 = 61,410

Case 3: Selecting 3 cards from one suit and 2 cards from another suit
In this case, we select 3 cards from one suit and 2 cards from another suit. Again, there are 4 suits to choose from and we can select any 2 suits for this case. The number of ways to select 3 cards from one suit and 2 cards from another suit is:

4C2 * 13C3 * 13C2 = 6 * 286 * 78 = 133,848

Case 4: Selecting 2 cards from one suit and 3 cards from another suit
In this case, we select 2 cards from one suit and 3 cards from another suit. We can choose any 2 suits out of the 4 available suits. The number of ways to select 2 cards from one suit and 3 cards from another suit is:

4C2 * 13C2 * 13C3 = 6 * 78 * 286 = 133,848

Case 5: Selecting 1 card from one suit and 4 cards from another suit
In this case, we select 1 card from one suit and 4 cards from another suit. We can choose any 2 suits out of the 4 available suits. The number of ways to select 1 card from one suit and 4 cards from another suit is:

4C2 * 13C1 * 13C4 = 6 * 13 * 715 = 61,410

Total Number of Ways
To find the total number of ways of selecting 5 cards from a deck of 52 cards such that one card of each suit is taken, we add up the results from all the cases:

28,561 + 61,410 + 133,848 + 133,848 + 61,410 = 419,077

Therefore, there are 419,077 different ways to select