Question

A debate club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this club including the selection of a captain (from among these 4 members) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is (A) 380 (B) 320 (C) 260 (D) 95

Short answer

380

Step-by-step solution

Calculate the number of teams with no boys

The number of ways to choose a team of 4 members from 6 girls (with no boys) is given by the combination formula, which is \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \). So, here it would be \( \binom{6}{4} \). After choosing the team, we select one captain from these 4 members, which can be done in 4 ways. Thus, the total number of ways for a team with no boys is \( \binom{6}{4} \times 4 \).

Calculate the number of teams with exactly one boy

The number of ways to choose a team of 3 girls from 6 and 1 boy from 4 is given by the product of combinations, which is \( \binom{6}{3} \times \binom{4}{1} \). After the team is chosen, we select one captain from these 4 members, which again can be done in 4 ways. Thus, the total number of ways for a team with exactly one boy is \( \binom{6}{3} \times \binom{4}{1} \times 4 \).

Calculate the total number of possible teams

The total number of ways to select the team is the sum of the number of ways to create teams with no boys and teams with exactly one boy. So the formula is the sum of the results from Step 1 and Step 2:\( (\binom{6}{4} \times 4) + (\binom{6}{3} \times \binom{4}{1} \times 4) \).

Compute the combinations and obtain the result

Using the combination formulas, we compute\( \binom{6}{4} = \frac{6!}{4!2!} = 15 \) and \( \binom{6}{3} \times \binom{4}{1} = \frac{6!}{3!3!} \times 4 = 20 \times 4 = 80 \). Multiplying by the 4 ways to choose a captain, the total number of ways is \( (15 \times 4) + (80 \times 4) = 60 + 320 = 380 \).

Key concepts

Combinations

Combinations refer to the selection of items from a larger set where order does not matter. When solving problems involving combinations, we use the formula \[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \],where \( n \) is the total number of items to choose from, \( r \) is the number of items to choose, and \( ! \) denotes factorial notation.

In the exercise, combinations are used to determine the possible ways of forming teams from a group of boys and girls. For example, the solution calculates the number of ways to select four girls out of six without considering the order using the formula \[ \binom{6}{4} \].The idea of combinations is crucial when it comes to group selection, lottery tickets, card games, and many other scenarios where the arrangement of the selected items doesn't affect the outcome.

Permutations

Permutations concern the arrangement of items where order is important. Different from combinations, every sequence of selected items is unique. The general formula for permutations is \[ nPr = \frac{n!}{(n-r)!} \], where again \( n \) is the total number of items and \( r \) is the number of items to arrange.

While the exercise deals with combinations, it is essential to understand the contrast with permutations. For instance, choosing a captain from the selected members involves permutation because the position of the captain is specific, and order matters. However, since there is only one position to fill, this simplifies to having the same number of permutations as the number of ways to choose a single captain.

Factorial Notation

Factorial notation, signified by an exclamation mark (\( ! \)), is a mathematical concept used to describe the product of an integer and all the integers below it down to one. For example, \( 5! \) (read as 'five factorial') is \( 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

Factorials are fundamental in both permutations and combinations. They are the building blocks of computing how many ways items can be selected or arranged. In the solution steps, factorial notation is used to simplify the combination formulas, such as in \[ \binom{6}{4} = \frac{6!}{4!2!} \],which reduces to a simple computation of factorials.

Binomial Coefficients

Binomial coefficients are the numbers that appear as coefficients in the expansion of a binomial raised to a power, such as in the binomial theorem. These coefficients can be represented using the same combination notation \[ \binom{n}{r} \].

Each binomial coefficient corresponds to the number of ways to choose \( r \) elements out of \( n \) elements, which is exactly the concept of combinations. The binomial coefficients have many applications in probability, statistics, and combinatorics. In our exercise, the binomial coefficients represent the different ways to select members for the debate club team. They reveal not just how many combinations exist but also the structure of potential groupings that occur when forming teams based on certain criteria, like having at most one boy on the team.