The binomial coefficient $\binom{24}{7}$, also known as "24 choose 7", is calculated using the formula:

$\binom{n}{r} = \frac{n!}{r!(n - r)!}$

Where:

• $n = 24$
• $r = 7$

Plugging the values into the formula:

$\binom{24}{7} = \frac{24!}{7!(24 - 7)!} = \frac{24!}{7!17!}$

This simplifies to:

$\binom{24}{7} = \frac{24 \times 23 \times 22 \times 21 \times 20 \times 19 \times 18}{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}$

Let's calculate the value:

$\binom{24}{7} = \frac{24 \times 23 \times 22 \times 21 \times 20 \times 19 \times 18}{5040}$

First, calculate the numerator:

$24 \times 23 = 552$ $552 \times 22 = 12144$ $12144 \times 21 = 255024$ $255024 \times 20 = 5100480$ $5100480 \times 19 = 96909120$ $96909120 \times 18 = 1744336416$

Now, divide by the denominator:

$\frac{1744336416}{5040} = 345471$

Therefore, $\binom{24}{7} = 345471$.

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Would you like more details or have any other questions on this?

Here are some related questions for you to explore:

1. What is the formula for calculating combinations?
2. How do binomial coefficients relate to Pascal's Triangle?
3. What is the difference between permutations and combinations?
4. How would you calculate $\binom{24}{0}$ or $\binom{24}{24}$?
5. What is the significance of binomial coefficients in probability theory?

Tip: To simplify large factorials in binomial coefficient calculations, always cancel out terms that appear in both the numerator and the denominator.