A club is going to send 4 of its 10 members to represent the club at a conference. How many different groups of 4 members can they send?

Calculate factorial of 10: To solve this problem, we need to calculate the number of combinations of 10 members taken 4 at a time. This is a combinatorial problem where order does not matter, and we can use the combination formula which is C(n, k) = n! / (k!(n-k)!), where n is the total number of items, k is the number of items to choose, and ＂!＂ denotes factorial.Calculate factorial of 4: First, we calculate the factorial of 10, which is 10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1.Calculate factorial of difference: Next, we calculate the factorial of 4, which is 4! = 4 × 3 × 2 × 1.Apply combination formula: Then, we calculate the factorial of the difference between 10 and 4, which is 6! = 6 × 5 × 4 × 3 × 2 × 1.Substitute factorial values: Now we can plug these values into the combination formula: C(10, 4) = 10! / (4! × (10-4)!).Simplify the equation: Substitute the factorial values into the formula: C(10, 4) = (10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1) / ((4 × 3 × 2 × 1) × (6 × 5 × 4 × 3 × 2 × 1)).Perform division and multiplication: We can simplify the equation by canceling out the common factors in the numerator and the denominator. The 6 × 5 × 4 × 3 × 2 × 1 in the denominator cancels with the same factors in the numerator, leaving us with: C(10, 4) = (10 × 9 × 8 × 7) / (4 × 3 × 2 × 1).Final result: Perform the division and multiplication: C(10, 4) = (10 × 9 × 8 × 7) / (24) = 90 × 7 = 630.Therefore, the club can send 630 different groups of 4 members to represent it at the conference.

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A club is going to send 4 of its 10 members to represent the club at a conference.
How many different groups of 4 members can they send?

A club is going to send $4$ of its $10$ members to represent the club at a conference. $\newline$ How many different groups of $4$ members can they send?

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Q. A club is going to send

4

of its

10

members to represent the club at a conference.

\newline

How many different groups of

4

members can they send?

Calculate factorial of $10$ : To solve this problem, we need to calculate the number of combinations of $10$ members taken $4$ at a time. This is a combinatorial problem where order does not matter, and we can use the combination formula which is $C(n, k) = \frac{n!}{k!(n-k)!}$ , where $n$ is the total number of items, $k$ is the number of items to choose, and “ $!$ ” denotes factorial.
Calculate factorial of $4$ : First, we calculate the factorial of $10$ , which is $10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$ .
Calculate factorial of difference: Next, we calculate the factorial of $4$ , which is $4! = 4 \times 3 \times 2 \times 1$ .
Apply combination formula: Then, we calculate the factorial of the difference between $10$ and $4$ , which is $6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1$ .
Substitute factorial values: Now we can plug these values into the combination formula: $C(10, 4) = \frac{10!}{4! \times (10-4)!}$ .
Simplify the equation: Substitute the factorial values into the formula: $C(10, 4) = \frac{10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{(4 \times 3 \times 2 \times 1) \times (6 \times 5 \times 4 \times 3 \times 2 \times 1)}$ .
Perform division and multiplication: We can simplify the equation by canceling out the common factors in the numerator and the denominator. The $6 \times 5 \times 4 \times 3 \times 2 \times 1$ in the denominator cancels with the same factors in the numerator, leaving us with: $C(10, 4) = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1}$ .
Final result: Perform the division and multiplication: $C(10, 4) = \frac{10 \times 9 \times 8 \times 7}{24} = 90 \times 7 = 630$ .
Final result: Perform the division and multiplication: $C(10, 4) = \frac{10 \times 9 \times 8 \times 7}{24} = 90 \times 7 = 630$ .Therefore, the club can send $630$ different groups of $4$ members to represent it at the conference.