Question

student painters, which specializes in painting the exterior of residential buildings, has eight people available to be organized into five - person and three - person teams. within a team, each person has a different job.
(a) using the generalized multiplication principle, determine how many ways a five - person team can be formed?
______ ways
(b) using the generalized multiplication principle, determine how many ways a three - person team can be formed?
______ ways
(c) using your results from parts (a) and (b), determine how many ways the company can form either a five - person team or a three - person team from the available people?
______ ways
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determine whether the statement is true or false. if it is true, explain why it is true. if it is false, give an example to show why it is false.
if there are six toppings available, then the number of different pizzas that can be made is (2^{6}), or 32, different pizzas.
(\bigcirc) true. this follows from the multiplication principle.
(\bigcirc) false. there are 6 different pizzas.
(\bigcirc) false. use the multiplication principle to conclude that there are (6\times6), or 36 different pizzas.
(\bigcirc) false. use the multiplication principle to conclude that there are (2\times6), or 12 different pizzas.
(\bigcirc) false. use the multiplication principle to conclude that there are (2\times2\times2\times2\times2\times2), or 64 different pizzas.

Explanation:

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First Problem (Parts a, b, c)

Step1: Calculate 5-person ordered teams

We use permutations (since each job is distinct): choose 5 people from 8 and arrange them.
$$P(8,5) = 8 \times 7 \times 6 \times 5 \times 4$$

Step2: Calculate 3-person ordered teams

Choose 3 people from 8 and arrange them (distinct jobs).
$$P(8,3) = 8 \times 7 \times 6$$

Step3: Sum the two results

"Or" means we add the number of ways for each team type.
$$P(8,5) + P(8,3)$$

For each topping, there are 2 choices: include it or exclude it. With 6 toppings, the multiplication principle gives $2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^6 = 64$ total pizza combinations. The original statement claims $2^5=32$, which is false.

Answer:

(a) $6720$ ways
(b) $336$ ways
(c) $7056$ ways

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Second Problem (True/False)