from a standard 52 card deck, how many 7 card hands have exactly 5 spades and 2 hearts?
Consider how many choices you have for each card drawn:
13*12*11*10*9 * 13*12
To determine the number of 7-card hands that have exactly 5 spades and 2 hearts from a standard 52-card deck, we can break down the problem into smaller steps:
Step 1: Calculate the number of ways to choose 5 spades out of the 13 spades in the deck.
This can be done using the combination formula C(n, r), where n is the total number of items and r is the number of items being chosen. In this case, there are 13 spades, so the calculation would be C(13, 5).
Step 2: Calculate the number of ways to choose 2 hearts out of the 13 hearts in the deck.
Similar to Step 1, we use the combination formula to calculate C(n, r), where n is the total number of items (13 hearts) and r is the number of items being chosen (2 hearts). So the calculation would be C(13, 2).
Step 3: Calculate the number of remaining cards to choose (7 - 5 - 2) from the remaining deck.
Since we have already chosen 5 spades and 2 hearts, the remaining cards will be 7 - 5 - 2 = 0 cards.
Step 4: Multiply the results of the three steps together to get the total number of 7-card hands.
Multiply the results from Step 1, Step 2, and Step 3 together to get the total number of 7-card hands with exactly 5 spades and 2 hearts.
The final answer would be the product of the three calculations:
C(13, 5) * C(13, 2) * C(36, 0).