Answer the questions below. \( \begin{array}{l}\text { (a) A company has } 33 \text { salespeople. A board member at the company asks for a list of the top } 4 \\ \text { salespeople, ranked in order of effectiveness. How many such rankings are possible? } \\ \text { (b) How many different committees of size } 4 \text { can be formed from } 14 \text { people? } \\ \text { (b) }\end{array} \)

Let's solve each part of the problem step by step. ### Part (a) We need to find the number of ways to rank the top 4 salespeople out of 33. This is a permutation problem because the order of selection matters (ranking). The formula for permutations is given by: \[ P(n, r) = \frac{n!}{(n - r)!} \] where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose. In this case, \( n = 33 \) and \( r = 4 \). Now, we can calculate \( P(33, 4) \): \[ P(33, 4) = \frac{33!}{(33 - 4)!} = \frac{33!}{29!} = 33 \times 32 \times 31 \times 30 \] Let's calculate this value. ### Part (b) For this part, we need to find the number of different committees of size 4 that can be formed from 14 people. This is a combination problem because the order of selection does not matter. The formula for combinations is given by: \[ C(n, r) = \frac{n!}{r!(n - r)!} \] where \( n \) is the total number of items, and \( r \) is the number of items to choose. In this case, \( n = 14 \) and \( r = 4 \). Now, we can calculate \( C(14, 4) \): \[ C(14, 4) = \frac{14!}{4!(14 - 4)!} = \frac{14!}{4! \times 10!} \] This can be simplified to: \[ C(14, 4) = \frac{14 \times 13 \times 12 \times 11}{4 \times 3 \times 2 \times 1} \] Now, let's calculate both values for parts (a) and (b). Calculate the value by following steps: - step0: Calculate: \(33\times 32\times 31\times 30\) - step1: Multiply the terms: \(1056\times 31\times 30\) - step2: Multiply the terms: \(32736\times 30\) - step3: Multiply the numbers: \(982080\) Calculate or simplify the expression \( (14*13*12*11)/(4*3*2*1) \). Calculate the value by following steps: - step0: Calculate: \(\frac{\left(14\times 13\times 12\times 11\right)}{\left(4\times 3\times 2\times 1\right)}\) - step1: Remove the parentheses: \(\frac{14\times 13\times 12\times 11}{4\times 3\times 2\times 1}\) - step2: Multiply the terms: \(\frac{24024}{4\times 3\times 2\times 1}\) - step3: Multiply the terms: \(\frac{24024}{24}\) - step4: Reduce the fraction: \(1001\) The calculations yield the following results: ### Part (a) The number of ways to rank the top 4 salespeople out of 33 is \( 982080 \). ### Part (b) The number of different committees of size 4 that can be formed from 14 people is \( 1001 \). In summary: - (a) There are \( 982080 \) possible rankings of the top 4 salespeople. - (b) There are \( 1001 \) different committees of size 4 that can be formed from 14 people.