Find the value of the binomial coefficient C(7, 4).

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Question: Find the value of the binomial coefficient C(7, 4).

Options:

Correct Answer: 35

Solution:

C(7, 4) = 7! / (4! * (7-4)!) = 7! / (4! * 3!) = (7*6*5)/(3*2*1) = 35.

Find the value of the binomial coefficient C(7, 4).

Find the value of the binomial coefficient C(7, 4).

Step 1: Understand that C(7, 4) is a way to choose 4 items from a total of 7 items.
Step 2: Use the formula for the binomial coefficient: C(n, k) = n! / (k! * (n-k)!). Here, n = 7 and k = 4.
Step 3: Substitute the values into the formula: C(7, 4) = 7! / (4! * (7-4)!).
Step 4: Calculate (7-4) which equals 3. So, we have C(7, 4) = 7! / (4! * 3!).
Step 5: Now, calculate the factorials: 7! = 7 * 6 * 5 * 4 * 3 * 2 * 1, 4! = 4 * 3 * 2 * 1, and 3! = 3 * 2 * 1.
Step 6: Substitute the factorials into the equation: C(7, 4) = 7! / (4! * 3!) = 7! / (24 * 6).
Step 7: Simplify the equation: C(7, 4) = 7! / 144.
Step 8: Calculate 7! = 5040. So, C(7, 4) = 5040 / 144.
Step 9: Divide 5040 by 144 to get 35.
Step 10: Therefore, the value of C(7, 4) is 35.

Binomial Coefficient – The binomial coefficient C(n, k) represents the number of ways to choose k elements from a set of n elements without regard to the order of selection.
Factorial Calculation – Understanding how to compute factorials is essential for calculating binomial coefficients.
Combinatorial Interpretation – The binomial coefficient can be interpreted in terms of combinations, which is a fundamental concept in combinatorics.