3.19: Password counting and team selection

0606/13/O/N/23 — Question 5 · 10 marks

A 5-character password is to be formed from the following 10 characters.
Letters: A, B, C, X, Y, Z
Symbols: ∗*, $\text{\$}, #\#, &\&
No character can be used more than once in any 5-character password.
(a)(i)
Find the number of passwords that can be formed.
[1]
(a)(ii)
Find the number of passwords that can be formed if the password has to contain at least one symbol.
[2]
(a)(iii)
Find the number of passwords that can be formed if the password has to start with two letters and end with two symbols.
[2]
(b)
A team of 88 people is to be chosen from 55 doctors, 44 teachers and 66 police officers.
Find how many possible teams have the same number of doctors as teachers.
[5]