The English alphabet contains 21 consonants and five vowels. How many strings of six lowercase letters of the English alphabet contain a) exactly one vowel? b) exactly two vowels? c) at least one vowel? d) at least two vowels?

Question

Hi Answer to each option is given as:

a) 122,523,030
b) 72,930,375
c) 223,149,655
d) 100,626,625
And I used the following approach to each option but answers don't match.
a) C(5,1)*C(21,5)*6!
b) C(5,2)*C(21,4)*6!
c) P(26,6)-P(21,6)
d) [C(5,2)*C(21,4)+C(5,3)*C(21,3)+C(5,4)*C(21,2)+C(5,5)*C(21,1)]*6!
Please check.

Answer 1

(a) Six letters with one vowel. We can place the vowel in any of the 6 positions in 5 * 6 ways and the remaining 5 positions can be filled in in 21⁵ ways by the consonants as they can be repeated also. So, total number of words possible

= 21⁵ * 5 * 6 = 122,523,030

(b) Two vowels will give (vowels can also be repeated)

21⁴ * 5 * 5 * 6C2  = 72,930,375

(c) 26⁶ - 21⁶ = 223,149,655

(d) (c) - (a) = 100,626,625

@Sahil-gupta All your answers are correct if letters were not allowed to be repeated. 

Comments

Thanks.
Got it.

---

What is mean by 6C2 in part b?