Let's solve these two problems step by step.

(a) Problem Analysis:

We are tasked with determining how many rankings are possible for the top 5 salespeople, ranked in order of effectiveness, from a group of 31 salespeople.

This is a permutation problem because the order in which the salespeople are ranked matters. The formula for permutations is:

$P(n, r) = \frac{n!}{(n-r)!}$

Where:

• $n = 31$ (total salespeople),
• $r = 5$ (top salespeople being ranked).

Solution for (a):

Substitute $n = 31$ and $r = 5$ into the formula:

$P(31, 5) = \frac{31!}{(31-5)!} = \frac{31!}{26!}$

Expanding the factorial:

$P(31, 5) = 31 \times 30 \times 29 \times 28 \times 27$

Now calculate:

$31 \times 30 = 930$ $930 \times 29 = 26,970$ $26,970 \times 28 = 755,160$ $755,160 \times 27 = 20,389,320$

Thus, the number of possible rankings is:

$\boxed{20,389,320}$

(b) Problem Analysis:

We are tasked with choosing 3 customers from a collection of 50 store customers. Here, the order does not matter. This is a combination problem.

The formula for combinations is:

$C(n, r) = \frac{n!}{r!(n-r)!}$

Where:

• $n = 50$ (total customers),
• $r = 3$ (customers to be chosen).

Solution for (b):

Substitute $n = 50$ and $r = 3$ into the formula:

$C(50, 3) = \frac{50!}{3!(50-3)!} = \frac{50 \times 49 \times 48}{3 \times 2 \times 1}$

Now calculate step by step:

1. Numerator: $50 \times 49 \times 48$: $50 \times 49 = 2,450$ $2,450 \times 48 = 117,600$

2. Denominator: $3 \times 2 \times 1 = 6$.

3. Divide: $\frac{117,600}{6} = 19,600$

Thus, the number of possible groups is:

$\boxed{19,600}$

Final Answers:

(a) The number of rankings for the top 5 salespeople is 20,389,320.
(b) The number of groups of 3 customers is 19,600.

Do you want further details or explanations for any step? Let me know!

Here are 5 related questions for practice:

1. How many ways can 4 people be ranked from a group of 10?
2. In how many ways can a team of 6 players be chosen from a group of 20?
3. How many ways can a president, vice-president, and secretary be chosen from 25 candidates?
4. From a group of 60 items, how many ways can 5 be selected?
5. How many permutations are there of arranging 7 people in 3 positions?

Tip: Always identify whether the problem involves permutations (order matters) or combinations (order does not matter) before applying formulas.