Product counting and estimating powers/roots
3 exam-style questions. Answers and the mark for each step are on the last page.
- Question 1
A code is made from two letters of the alphabet (A to Z) followed by three digits (0 to 9).
(a) How many different codes are possible?
(b) How many codes are possible if the two letters must be different?
- Question 2
A three-digit number has no repeated digits and cannot start with 0.
(a) How many such three-digit numbers are even?
- Question 3
(a) How many six-digit positive integers can be made using each of the digits 0, 1, 2, 3, 4 and 5 exactly once, if the integer must be divisible by 4? A six-digit integer cannot begin with 0.
Answers and marks
Question 1
(a)
- M1 Multiplying the choices, such as .
- A1 The correct answer, .
(b)
- M1 Using for the letters.
- A1 The correct answer, .
Question 2
(a)
- P1 Splitting into cases: last digit 0, and last digit 2, 4, 6 or 8.
- P1 Counting the last-digit-0 case correctly: 72.
- P1 Counting the other case correctly: .
- A1 The correct answer, .
Question 3
(a)
- P1 Establishing or an equivalent valid method.
- P1 Establishing or an equivalent valid method.
- P1 Establishing or an equivalent valid method.
- A1 Correct answer: