Solved on Nov 27, 2023

(a) Find the number of 5-letter passwords with no repeated lowercase letters from 26 alphabet letters. (b) A company has 33 salespeople. Find the number of possible rankings of the top 3 salespeople.
(a) $\binom{26}{5}$ (b) $\binom{33}{3}$

STEP 1

Assumptions1. For the computer password, no letter may be repeated and only the lower case of a letter may be used. There are26 letters in the alphabet. . For the salespeople ranking, there are33 salespeople in total and we need to find the number of ways to rank the top3.

STEP 2

For part (a), we need to find the number of ways to choose5 letters from26 without repetition. This is a permutation problem, and the formula for permutations is $(n, r) = \frac{n!}{(n-r)!}$ where n is the total number of items, r is the number of items to choose, and "!" denotes factorial.

STEP 3

Now, plug in the given values for n and r to calculate the number of possible passwords.
$(26,5) = \frac{26!}{(26-5)!}$

STEP 4

Calculate the number of possible passwords.
$(26,) = \frac{26!}{21!}$

STEP 5

implify the factorial expression to get the number of possible passwords.
$(26,5) =26 \times25 \times24 \times23 \times22 =7,893,600$

STEP 6

For part (b), we need to find the number of ways to rank the top3 salespeople out of33. This is also a permutation problem, and we can use the same formula as in part (a).

STEP 7

Now, plug in the given values for n and r to calculate the number of possible rankings.
$(33,3) = \frac{33!}{(33-3)!}$

STEP 8

Calculate the number of possible rankings.
$(33,3) = \frac{33!}{30!}$

STEP 9

implify the factorial expression to get the number of possible rankings.
$(33,3) =33 \times32 \times31 =32,856$ So, there are7,893,600 possible5-letter passwords and32,856 possible rankings of the top3 salespeople.

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(a) Find the number of 5-letter passwords with no repeated lowercase letters from 26 alphabet letters. (b) A company has 33 salespeople. Find the number of possible rankings of the top 3 salespeople.
(a) $\binom{26}{5}$ (b) $\binom{33}{3}$

STEP 1

Assumptions1. For the computer password, no letter may be repeated and only the lower case of a letter may be used. There are26 letters in the alphabet. . For the salespeople ranking, there are33 salespeople in total and we need to find the number of ways to rank the top3.

STEP 2

For part (a), we need to find the number of ways to choose5 letters from26 without repetition. This is a permutation problem, and the formula for permutations is $(n, r) = \frac{n!}{(n-r)!}$ where n is the total number of items, r is the number of items to choose, and "!" denotes factorial.

STEP 3

Now, plug in the given values for n and r to calculate the number of possible passwords.
$(26,5) = \frac{26!}{(26-5)!}$

STEP 4

Calculate the number of possible passwords.
$(26,) = \frac{26!}{21!}$

STEP 5

implify the factorial expression to get the number of possible passwords.
$(26,5) =26 \times25 \times24 \times23 \times22 =7,893,600$

STEP 6

For part (b), we need to find the number of ways to rank the top3 salespeople out of33. This is also a permutation problem, and we can use the same formula as in part (a).

STEP 7

Now, plug in the given values for n and r to calculate the number of possible rankings.
$(33,3) = \frac{33!}{(33-3)!}$

STEP 8

Calculate the number of possible rankings.
$(33,3) = \frac{33!}{30!}$

STEP 9

implify the factorial expression to get the number of possible rankings.
$(33,3) =33 \times32 \times31 =32,856$ So, there are7,893,600 possible5-letter passwords and32,856 possible rankings of the top3 salespeople.

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(a) Find the number of 5-letter passwords with no repeated lowercase letters from 26 alphabet letters. (b) A company has 33 salespeople. Find the number of possible rankings of the top 3 salespeople.(a) (265)\binom{26}{5}(526​) (b) (333)\binom{33}{3}(333​)

STEP 1

Assumptions1. For the computer password, no letter may be repeated and only the lower case of a letter may be used. There are26 letters in the alphabet. . For the salespeople ranking, there are33 salespeople in total and we need to find the number of ways to rank the top3.

STEP 2

STEP 3

Now, plug in the given values for n and r to calculate the number of possible passwords.(26,5)=26!(26−5)!(26,5) = \frac{26!}{(26-5)!}(26,5)=(26−5)!26!​

STEP 4

Calculate the number of possible passwords.(26,)=26!21!(26,) = \frac{26!}{21!}(26,)=21!26!​

STEP 5

implify the factorial expression to get the number of possible passwords.(26,5)=26×25×24×23×22=7,893,600(26,5) =26 \times25 \times24 \times23 \times22 =7,893,600(26,5)=26×25×24×23×22=7,893,600

STEP 6

For part (b), we need to find the number of ways to rank the top3 salespeople out of33. This is also a permutation problem, and we can use the same formula as in part (a).

STEP 7

Now, plug in the given values for n and r to calculate the number of possible rankings.(33,3)=33!(33−3)!(33,3) = \frac{33!}{(33-3)!}(33,3)=(33−3)!33!​

STEP 8

Calculate the number of possible rankings.(33,3)=33!30!(33,3) = \frac{33!}{30!}(33,3)=30!33!​

STEP 9

implify the factorial expression to get the number of possible rankings.(33,3)=33×32×31=32,856(33,3) =33 \times32 \times31 =32,856(33,3)=33×32×31=32,856So, there are7,893,600 possible5-letter passwords and32,856 possible rankings of the top3 salespeople.

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

(a) Find the number of 5-letter passwords with no repeated lowercase letters from 26 alphabet letters. (b) A company has 33 salespeople. Find the number of possible rankings of the top 3 salespeople.
(a) $\binom{26}{5}$ (b) $\binom{33}{3}$

Now, plug in the given values for n and r to calculate the number of possible passwords.
$(26,5) = \frac{26!}{(26-5)!}$

Calculate the number of possible passwords.
$(26,) = \frac{26!}{21!}$

implify the factorial expression to get the number of possible passwords.
$(26,5) =26 \times25 \times24 \times23 \times22 =7,893,600$

Now, plug in the given values for n and r to calculate the number of possible rankings.
$(33,3) = \frac{33!}{(33-3)!}$

Calculate the number of possible rankings.
$(33,3) = \frac{33!}{30!}$

implify the factorial expression to get the number of possible rankings.
$(33,3) =33 \times32 \times31 =32,856$ So, there are7,893,600 possible5-letter passwords and32,856 possible rankings of the top3 salespeople.