Suppose we want to choose 5 letters, without replacement, from 15 distinct letters.
(a) How many ways can this be done, if the order of the choices matters?

(b) How many ways can this be done, if the order of the choices does not matter?

Question

Suppose we want to choose 5 letters, without replacement, from 15 distinct letters.
(a) How many ways can this be done, if the order of the choices matters?

(b) How many ways can this be done, if the order of the choices does not matter?

UpStudy AI · Accepted Answer

To solve this problem, we can use the concept of permutations and combinations.

(a) If the order of the choices matters, we are looking for the number of permutations of 5 letters chosen from 15 distinct letters. The formula for permutations is given by:

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

where:
- $$ n $$ is the total number of items (15 letters in this case),
- $$ r $$ is the number of items to be chosen (5 letters in this case).

Substitute the values into the formula to find the number of ways to choose 5 letters without replacement, considering the order of the choices.

(b) If the order of the choices does not matter, we are looking for the number of combinations of 5 letters chosen from 15 distinct letters. The formula for combinations is given by:

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

where:
- $$ n $$ is the total number of items (15 letters in this case),
- $$ r $$ is the number of items to be chosen (5 letters in this case).

Substitute the values into the formula to find the number of ways to choose 5 letters without replacement, considering that the order of the choices does not matter.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{15!}{\left(15-5\right)!}$$
- step1: Subtract the numbers:
  $$\frac{15!}{10!}$$
- step2: Expand the expression:
  $$\frac{15\times 14\times 13\times 12\times 11\times 10!}{10!}$$
- step3: Simplify:
  $$15\times 14\times 13\times 12\times 11$$
- step4: Multiply the terms:
  $$210\times 13\times 12\times 11$$
- step5: Multiply the terms:
  $$2730\times 12\times 11$$
- step6: Multiply the terms:
  $$32760\times 11$$
- step7: Multiply the numbers:
  $$360360$$
(a) If the order of the choices matters, there are 360,360 ways to choose 5 letters without replacement from 15 distinct letters.

(b) If the order of the choices does not matter, there are 3,003 ways to choose 5 letters without replacement from 15 distinct letters.
(a) If the order matters, there are 360,360 ways to choose 5 letters. (b) If the order does not matter, there are 3,003 ways to choose 5 letters.

Suppose we want to choose 5 letters, without replacement, from 15 distinct letters.
(a) How many ways can this be done, if the order of the choices matters?

(b) How many ways can this be done, if the order of the choices does not matter?

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Latest Probability Questions

Suppose we want to choose 5 letters, without replacement, from 15 distinct letters. (a) How many ways can this be done, if the order of the choices matters? (b) How many ways can this be done, if the order of the choices does not matter?

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Latest Probability Questions

Suppose we want to choose 5 letters, without replacement, from 15 distinct letters.
(a) How many ways can this be done, if the order of the choices matters?

(b) How many ways can this be done, if the order of the choices does not matter?