Question

In how many ways can the letters of the word COMBINE be arranged so as to begin and end with a vowel? Also find the number of words that can be formed without changing the relative order of the vowels and consonants.

Answer 1

Total $=7$ out of which 3 vowels 4 consonants
(i) ${}^3{P}_2\times 5!=6\times 120=720$
(ii) The three vowels shall continue to occupy the $2^{nd},5^{th}$ and $7^{th}$ position only. In other words, these three vowels can be arranged in these three places in $3!=6$ ways. Similarly the four consonants can be arranged in $4!=24$ ways. Thus the total number of letters is $6\times 24=144$ in which the relative order is maintained.