4. In how many ways can 4 letters be arranged to form ordered codes of 3 letters...

Questions

October 2022 1 1 Report

4. In how many ways can 4 letters be arranged to form ordered codes of 3 letters if repetition is not allowed?
5. In how many ways can 8 numbers be arranged to form all possible 4-digit numbers?

Answers & Comments

NayeonTheCuteBunny

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$\:$ 4. In how many ways can 4 letters be arranged to form ordered codes of 3 letters if repetition is not allowed?

$\:$ 5. In how many ways can 8 numbers be arranged to form all possible 4-digit numbers?

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$\:$ $\rm {4. \: 24 \: ways}$

$\:$ $\rm {5. \: 4 \: 096 \: ways}$

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$\large\boxed{\mathbb\red{NUMBER \: 4.}}$

Given:

P = ?
n = 4
r = 3

» Since repetition is not allowed, we will use the formula:

$\tt{_{n}P_r}$ = $\tt {\frac{n!}{(n - r)!}}$

» Substitute the givens on the formula.

$\tt{_{4}P_3}$ = $\frac{4!}{(4 - 3)!}$

$\tt{_{4}P_3}$ = $\frac{4!}{1!}$

$\tt{_{4}P_3}$ = $\frac{4 × 3 × 2 × \cancel {1!}}{ \cancel {1!}}$

$\tt{_{4}P_3}$ = $\frac{4 × 3 × 2}{1}$

$\tt{_{4}P_3}$ = 24

Therefore, there are 24 ways to arrange the 4 letters to form ordered codes of 3 letters without repetition.

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$\large\boxed{\mathbb\red{NUMBER \: 5.}}$

Given:

P = ?
n = 8
r = 4

» Since it's asking how many ways can 8 numbers be arranged to form all possible 4-digit numbers, which is repetition or replacement is allowed, we will use the formula: