A code has four characters. The first character is one of the letters A, B, C, D, E, or F. Each of the other three characters is a digit from $0$ to $9$, and a digit may be used more than once. Find the number of possible codes.
Question HKC78-AAHL-CNT-01
[Maximum mark: 4]
A code has four characters. The first character is one of the letters A, B, C, D, E, or F. Each of the other three characters is a digit from $0$ to $9$, and a digit may be used more than once. Find the number of possible codes.
Mark Scheme
There are $6$ choices for the letter. [1]
Each of the three digits has $10$ choices, and repetition is allowed, so those positions contribute $10^{3}$. [2]
Total: $6 times 10^{3} = 6000$. [1]
Total: 4 marks
A committee of $3$ people is chosen from $5$ teachers and $4$ students. Find the number of committees that include at least one student.
Question HKC78-AAHL-CNT-02
[Maximum mark: 6]
A committee of $3$ people is chosen from $5$ teachers and $4$ students. Find the number of committees that include at least one student.
Mark Scheme
The number of committees with no restriction is $dbinom{9}{3} = 84$. [1]
The number with no student is $dbinom{5}{3} = 10$. [1]
At least one student: $84 – 10 = 74$. [2]
Check by cases: exactly one student $dbinom{4}{1}dbinom{5}{2} = 40$, exactly two $dbinom{4}{2}dbinom{5}{1} = 30$, exactly three $dbinom{4}{3} = 4$. Sum $40+30+4 = 74$. [2]
Total: 6 marks
