Chapter 15.5, Problem 20WE

To determine

To calculate: The number of 3-letter code if at least one of the letters is vowels.

Answer to Problem 20WE

The number of 3-letter code if at least one of the letters is vowelsare $2855$ .

Explanation of Solution

Given information:

3-letter code is to be formed and at least one of letter should be vowel.

Formula used:

Fundamental counting principle states that if there are m ways to make a selection and n ways to make a second selection then there are $m\times n$ number of ways in which two selection can be made.

Calculation:

Consider the provided information that 3-letter code is to be formed and at least one of letter should be vowel.

There are 5 vowels and 21 consonants.

Let 3-letter code wordbe $\boxed{}\boxed{}\boxed{}$ .

Case I: Only 1 vowel is there.

To select the first letter 5 choices are there among vowel. $\boxed{5}\boxed{}\boxed{}$ .

The remaining two boxes can be consonants and each box has 21 possibilities $\boxed{5}\boxed{21}\boxed{21}$ .

Recall that Fundamental counting principle states that if there are m ways to make a selection and n ways to make a second selection then there are $m\times n$ number of ways in which two selection can be made.

Therefore, the number of ways to select are,

  $5\times 21\times 21=2205$

Case II: Only 2 vowels are there.

To select the first letter 5 choices are there among vowel. $\boxed{5}\boxed{}\boxed{}$ .

To select the second letter 5 choices are there among vowel. $\boxed{5}\boxed{5}\boxed{}$ .

The remaining box can be consonant and has 21 possibilities $\boxed{5}\boxed{5}\boxed{21}$ .

Recall that Fundamental counting principle states that if there are m ways to make a selection and n ways to make a second selection then there are $m\times n$ number of ways in which two selection can be made.

Therefore, the number of ways to select are,

  $5\times 5\times 21=525$

Case III: All 3 vowels are there.

To select the first letter 5 choices are there among vowel. $\boxed{5}\boxed{}\boxed{}$ .

To select the second letter 5 choices are there among vowel. $\boxed{5}\boxed{5}\boxed{}$ .

To select the third letter 5 choices are there among vowel. $\boxed{5}\boxed{5}\boxed{5}$ .

Recall that Fundamental counting principle states that if there are m ways to make a selection and n ways to make a second selection then there are $m\times n$ number of ways in which two selection can be made.

Therefore, the number of ways to select are,

  $5\times 5\times 5=125$

Therefore, total number of 3-letter codes are,

  $125+525+2205=2855$

Thus, the number of 3-letter code if at least one of the letters is vowels are $2855$ .