Pregunta
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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?

Ask by Patel Marsh. in the United States
Mar 15,2025

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(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.

Solución

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 items from distinct items, where order matters, is given by the formula:
In this case, and . Therefore, we can calculate:
This simplifies to:
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 items from distinct items, where order does not matter, is given by the formula:
In this case, and . Therefore, we can calculate:
This simplifies to:
Now, let’s calculate both values.
I’ll perform the calculations for both parts.
Calculate the value by following steps:
  • step0: Calculate:
  • step1: Multiply the terms:
  • step2: Multiply the terms:
  • step3: Multiply the numbers:

    Calculate or simplify the expression .
    Calculate the value by following steps:
  • step0: Calculate:
  • step1: Remove the parentheses:
  • step2: Multiply the terms:
  • step3: Multiply the terms:
  • step4: Reduce the fraction:

Final Results

(a) When the order of the choices is relevant, the number of ways to choose 4 letters from 17 distinct letters is:
(b) When the order of the choices is not relevant, the number of ways to choose 4 letters from 17 distinct letters is:
Thus, the answers are:
  • (a) 57120
  • (b) 2380

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Bonus Knowledge

To solve part (a), where order matters, we use permutations. The number of ways to choose 4 letters from 17 distinct letters is given by the formula for permutations:
Here, is 17 and is 4. Thus,
So, there are 57,120 ways to choose 4 letters if the order of the choices is relevant.
For part (b), where order does not matter, we use combinations. The number of ways to choose 4 letters from 17 is given by the formula for combinations:
Thus,
Therefore, there are 2,380 ways to choose 4 letters without caring about the order.

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