Problem 49 - Algebra and Trigonometry

Question

A student dance committee is to be formed consisting of 2 boys and 3 girls. If the membership is to be chosen from 4 boys and 8 girls, how many different committees are possible?

Step-by-Step Solution

Answer

336

Step 1: Determine the number of ways to choose the boys

To form the committee, 2 boys need to be chosen from a total of 4 boys. The number of combinations in which this can be done is given by the binomial coefficient \(\binom{4}{2}\). This can be calculated using the formula for combinations \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).

Step 2: Calculate \(\binom{4}{2}\)

Calculate the number of ways to choose 2 boys from 4: \(\binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6\).

Step 3: Determine the number of ways to choose the girls

Next, 3 girls need to be chosen from a total of 8 girls. The number of combinations here is given by \(\binom{8}{3}\).

Step 4: Calculate \(\binom{8}{3}\)

Calculate the number of ways to choose 3 girls from 8: \(\binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56\).

Step 5: Combine the two results

The total number of different committees can be found by multiplying the number of ways to choose the boys and the number of ways to choose the girls. This is because each combination of boys can be paired with any combination of girls.

Step 6: Final calculation

Multiply the results from Steps 2 and 4: \(\binom{4}{2} \times \binom{8}{3} = 6 \times 56 = 336\).

Key Concepts

binomial coefficient

In combinatorics, the binomial coefficient, often denoted as \(\binom{n}{k}\), represents the number of ways to choose \(k\) elements from a set of \(n\) elements without considering the order of selection. The formula for the binomial coefficient is \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\). Here’s what each term means:

• \(n!\): the factorial of \(n\), representing the product of all integers from 1 to \(n\)
• \(k!\): the factorial of \(k\)
• \((n-k)!\): the factorial of \(n-k\)

The binomial coefficient is crucial in calculating combinations and is extensively used in probability and statistics.

combinations

Combinations are fundamental in finding the total number of ways to create groups or teams from a larger set.
Multiply both sets of combinations to find the final number of unique groups: \(6 \times 56 = 336\). This multiplicative principle works because each combination of boys can form a unique committee with each combination of girls.

factorial

The factorial of a number \(n\) (denoted as \(n!\)) is the product of all positive integers less than or equal to \(n\). For example, \(4! = 4 \times 3 \times 2 \times 1 = 24\). Factorials are key when calculating the binomial coefficient:

• The formula \(\binom{4}{2} \) includes \(4!\), \(2!\), and \( (4-2)! = 2!\)

Using these, the coefficient \(\binom{4}{2} = \frac{4!}{2!2!} = \frac{24}{2 \times 2} = 6\). Factorials grow rapidly with larger \(n\) . Factorial calculations are essential in solving combination problems in combinatorics to determine the number of ways to select items from a set without considering the order of selection.