Question

7 Find the number of ways all 10 letters of the word COPENHAGEN can be arranged so that (i) the vowels (A, E, O) are together and the consonants (C, G, H, N, P) are together,

Answer 1

4320 ways.

Explanation:

Question asked:

Find the number of ways all 10 letters of the word COPENHAGEN can be arranged so that (i) the vowels (A, E, O) are together and the consonants (C, G, H, N, P) are together,

Solution:

By using Permutation formula:

`^(n) P_(r) \ =(n!)/((n-r)!)`

`''n'' \ is\ the\ number\ of\ letters\ taking\''r'' at\ a\ time.`

CGHNP AEO EN

Total number of letters = 10

Let consonant (CGHNP) = C

And vowel (AEO) = V

Now we have only four letters CVEN

We can arrange this 4 letters in =
`^(4) P_(4) \ ways\\ \\`

`=(4!)/((4-4!)) \\ \\ =(4!)/((0!))\\ \\ =4*3*2*1=24\ ways`

Consonants having 5 letters arrange themselves in =
`^(5) P_(5) \ ways\\ \\`

`=(5!)/((5-5)!) \\ \\ =(5*4*3*2*1)/(0!) \\ \\ =120\ ways`

Vowels having 3 letters arrange themselves in =
`^(3) P_(3) \ ways\\ \\`

=
`=(3!)/((3-3)!) \\ \\ 3*2*1=6 \ ways`

Repeated letter :-

E = 2 times in
`^(2) P_(2) \ ways=2\ ways`

N = 2 times in 2 ways

Total arrangements of repeated letters = 2
`*` 2 = 4 ways

Total number of ways =
`(24*120*6)/(Repated\ letters\ arrangements)`

=
`(17280)/(4) =4320\ ways`

Therefore, the number of ways all 10 letters of the word can be arranged in 4320 ways.