Question

How many 7 card hands can be dealt from a deck of 40 cards?

Answer 1

Final answer:

There are 658,008,014,430 ways to deal a 7-card hand from a deck of 40 cards.

Step-by-step explanation:

To find the number of ways to choose a 7-card hand from a deck of 40 cards, you can use the combination formula. The formula for combinations is given by:

`\[ C(n, k) = (n!)/(k!(n-k)!) \]`

where
`\( n! \)` denotes the factorial of
`\( n \)`.

In this case, you want to find the number of ways to choose a 7-card hand from a deck of 40 cards, so
`\( n = 40 \)` and
`\( k = 7 \)`. Therefore, the calculation is:

`\[ C(40, 7) = (40!)/(7!(40-7)!) \]`

Now, compute the factorials and simplify the expression to find the number of 7-card hands. You can use a calculator or software with factorial function support to perform the calculation.

`\[ C(40, 7) = (40!)/(7! \cdot 33!) \]`

`\[ C(40, 7) = (40 \cdot 39 \cdot 38 \cdot 37 \cdot 36 \cdot 35 \cdot 34)/(7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1) \]`

`\[ C(40, 7) = 658,008,014,430 \]`

Therefore, there are
`\( 658,008,014,430 \)` ways to choose a 7-card hand from a deck of 40 cards.