Solution: 2014 Winter Final - 1
Author: Michiel SmidQuestion
A password consists of 28 characters, each character being a lowercase letter. A password must
contain exactly one vowel (i.e., a, e, i, o, or u).
How many passwords are there?
(a)
$28 \cdot 5 \cdot 26^{27}$
(b)
$5 \cdot 21^{27}$
(c)
$28 \cdot 5 \cdot 27^{21}$
(d)
$28 \cdot 5 \cdot 21^{27}$
Solution
The vowel has 5 possible outcomes$ { a,e,i,o,u } $: $ 5 $
It must choose 1 of the 28 character positions: $ 28 $
The remaining 27 lowercasebconsonants each have 21 possible outcomes
$ { b,c,d,f,g,h,j,k,l,m,n,p,q,r,s,t,v,w,x,y,z } $: $ 21^{27} $
The total number of passwords is the product of these outcomes: $ 5 \times 28 \times 21^{27} $