Answer to Question #346612 in Discrete Mathematics for Bryan joy Barrientos

Question

2. How many 4 letter words can be created, if repetitions are allowed?

3. How many three digit numbers can be formed?

4. How many words (of any number of letters) can be formed from GAME?

5. How many 3-digit, 4-digit, or 5-digit numbers can be made using the digits of 46723819?

6. How many numbers between 999 and 9999 are divisible by 5 and have no repeated digits?

7. How many ways can you order the letters in KEYBOARD if K and Y must always be kept together?

8. How many ways can 4 rock, 5 pop, and 6 classical albums be ordered if all albums of the same genre must be kept together?

9. In how many ways can the letters from the word EDITOR be arranged if vowels and consonants alternate positions?

10. If 8 boys and 2 girls must stand in line for a picture, how many line-up’s will have the girls separated from each other?

Answer 1

2. If repetition is allowed there are 26 possibilities for each letter. Therefore, there are $26^4=456976$ words.

3. If repetition is allowed there are $9(10)(10)=900$ numbers.

If repetition is not allowed there are $9(9)(8)=648$ numbers.

4.

$$P(4,1)+P(4,2)+P(4,3)+P(4,4)$$ $$=4+12+24+24=64$$

There are $64$ words.

5.

$$P(8,3)+P(8,4)+P(8,5)$$ $$=336+1680+6720=8736$$

If repetition is not allowed there are $8736$ numbers.

If repetition is allowed there are $8^3+8^4+8^5=37376$ numbers.

6.

Odd numbers divisible by 5

$$8(8)(7)=448$$

Even numbers divisible by 5

$$9(8)(7)=504$$ $$448+504=952$$

952 numbers

7.

$$2!(7)!=10080$$

8.

$$4!(5!)(6!)(3!)=12441600$$

9. The word ‘EDITOR’ is a six-letter word, having three vowels (e, i, o) and three consonants (d, t, r). So, there is no repetition of letters - neither vowels nor consonants.

A vowel is at the first place

$$3!(3!)=36$$

A consonant is at the first place

$$3!(3!)=36$$

The letters from the word EDITOR can be arranged in $36+36=72$ ways.

10. Two girls are together

$$2!(9!)=725760$$

There are

$$10!=3628800$$

possible arrangements.

Two girls are not together

$$3628800-725760=2903040$$