Suppose we want to choose 4 letters, without replacement, from 17 distinct letters.
(If necessary, consult a list of formulas.)

(b) If the order of the choices is not relevant, how many ways can this be done?

Ask by Patel Marsh. in the United States

Mar 15,2025

Question

Suppose we want to choose 4 letters, without replacement, from 17 distinct letters.
(If necessary, consult a list of formulas.)

(b) If the order of the choices is not relevant, how many ways can this be done?

Ask by Patel Marsh. in the United States

Mar 15,2025

UpStudy AI · Accepted Answer

To solve the problem of choosing 4 letters from 17 distinct letters, we will consider two scenarios: when the order of the choices is relevant and when it is not.

### (a) When the order of the choices is relevant

When the order matters, we are dealing with permutations. The number of ways to choose $$ r $$ items from $$ n $$ distinct items, where order matters, is given by the formula:

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

In this case, $$ n = 17 $$ and $$ r = 4 $$. Therefore, we can calculate:

$$
P(17, 4) = \frac{17!}{(17 - 4)!} = \frac{17!}{13!}
$$

This simplifies to:

$$
P(17, 4) = 17 \times 16 \times 15 \times 14
$$

Now, let's calculate this value.

### (b) When the order of the choices is not relevant

When the order does not matter, we are dealing with combinations. The number of ways to choose $$ r $$ items from $$ n $$ distinct items, where order does not matter, is given by the formula:

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

In this case, $$ n = 17 $$ and $$ r = 4 $$. Therefore, we can calculate:

$$
C(17, 4) = \frac{17!}{4!(17 - 4)!} = \frac{17!}{4! \cdot 13!}
$$

This simplifies to:

$$
C(17, 4) = \frac{17 \times 16 \times 15 \times 14}{4!}
$$

Now, let's calculate both values.

I'll perform the calculations for both parts.
Calculate the value by following steps:
- step0: Calculate:
  $$17\times 16\times 15\times 14$$
- step1: Multiply the terms:
  $$272\times 15\times 14$$
- step2: Multiply the terms:
  $$4080\times 14$$
- step3: Multiply the numbers:
  $$57120$$
Calculate or simplify the expression $$ (17 * 16 * 15 * 14) / (4 * 3 * 2 * 1) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(17\times 16\times 15\times 14\right)}{\left(4\times 3\times 2\times 1\right)}$$
- step1: Remove the parentheses:
  $$\frac{17\times 16\times 15\times 14}{4\times 3\times 2\times 1}$$
- step2: Multiply the terms:
  $$\frac{57120}{4\times 3\times 2\times 1}$$
- step3: Multiply the terms:
  $$\frac{57120}{24}$$
- step4: Reduce the fraction:
  $$2380$$
### Final Results

(a) When the order of the choices is relevant, the number of ways to choose 4 letters from 17 distinct letters is:

$$
P(17, 4) = 57120
$$

(b) When the order of the choices is not relevant, the number of ways to choose 4 letters from 17 distinct letters is:

$$
C(17, 4) = 2380
$$

Thus, the answers are:
- (a) 57120
- (b) 2380
(a) There are 57,120 ways to choose 4 letters from 17 distinct letters when the order matters. (b) There are 2,380 ways to choose 4 letters from 17 distinct letters when the order does not matter.

Suppose we want to choose 4 letters, without replacement, from 17 distinct letters.
(If necessary, consult a list of formulas.)

(b) If the order of the choices is not relevant, how many ways can this be done?

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Suppose we want to choose 4 letters, without replacement, from 17 distinct letters. (If necessary, consult a list of formulas.) (b) If the order of the choices is not relevant, how many ways can this be done?

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Latest Probability Questions

Suppose we want to choose 4 letters, without replacement, from 17 distinct letters.
(If necessary, consult a list of formulas.)

(b) If the order of the choices is not relevant, how many ways can this be done?