Pregunta
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?
(If necessary, consult a list of formulas.)
Ask by Patel Marsh. in the United States
Mar 15,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
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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:
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
Respondido por UpStudy AI y revisado por un tutor profesional
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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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