Let's solve the problem of determining in how many ways 50 cards can be chosen from a standard deck of 52 cards.
1. Understanding the Problem:
- We have a standard deck of 52 cards.
- We want to choose 50 cards from this deck.
- This is a combinatorial problem, where we need to find the number of combinations (ways to choose a subset) of 50 cards from 52 cards.
2. Combinatorial Formula:
- The number of ways to choose [tex]\( k \)[/tex] items from [tex]\( n \)[/tex] items without regard to order is given by the binomial coefficient, denoted as [tex]\( \binom{n}{k} \)[/tex].
- The formula for the binomial coefficient is:
[tex]\[
\binom{n}{k} = \frac{n!}{k!(n - k)!}
\][/tex]
- Here, [tex]\( n = 52 \)[/tex] (total cards) and [tex]\( k = 50 \)[/tex] (cards to choose).
3. Calculating the Binomial Coefficient:
- Plugging the values into the formula, we get:
[tex]\[
\binom{52}{50} = \frac{52!}{50!(52 - 50)!} = \frac{52!}{50! \cdot 2!}
\][/tex]
4. Result:
- Without manually computing the factorials (as it’s quite cumbersome), the number of ways to choose 50 cards from a deck of 52 cards is represented by the binomial coefficient [tex]\( \binom{52}{50} \)[/tex].
- The computed value for this coefficient is 1326.
Therefore, the number of ways to choose 50 cards from a standard deck of 52 cards is 1326.
