9 3-digit integers are made using three different digits.
The digits are chosen from \(\quad 1, 2, 3, 4, 5, 6, 7, 8, 9\)
How many more odd integers than even integers can be made? [3 marks]
Mark scheme
Answer
Mark
Comments
Alternative method 1: by considering the third digit
\(8 \times 7 \times 5\) or 280 or \(8 \times 7 \times 4\) or 224
M1
\(8 \times 7 \times 5\) or 280 and \(8 \times 7 \times 4\) or 224 or \(8 \times 7 \times (5 - 4)\)
M1dep
56
A1
SC1 81
Alternative method 2: splitting the total in the correct ratio
\(9 \times 8 \times 7\)
M1
\(\dfrac{504}{9}\) or 280 and 224
M1dep
56
A1
SC1 81
Additional guidance
Up to M1 may be awarded for correct work with no answer or incorrect answer, even if this is seen amongst multiple attempts
Listing outcomes with incorrect answer (could still gain M1 from above if part of multiple attempts)
M0A0
56 from clearly incorrect working
M0A0
56 with no incorrect working but some omitted working – this could come from listing groups of outcomes but missing some groups out (that give the same answers so cancel)
What is the greatest possible number of integers in the set that are multiples of 5? [3 marks]
Mark scheme
Answer
Mark
Comments
480
B3
B2 \(3 \times 8 \times 10 \times 2\) with at least three correct B1 \(3 \times 8 \times 10 \times 2\) with at least two correct or 3, 8, 10 and 2 identified with no others SC2 \(\dfrac{480}{10\,000}\) or \(\dfrac{1}{480}\) SC1 2400 or 720 or 189
Additional guidance
eg1 \(3 \times 8 \times 10 \times 10\) product of exactly four numbers eg2 \(6 \times 7 \times 10 \times 2\) product of exactly four numbers eg3 \(3 \times 8 \times 10\) product of only three numbers eg4 \(24 \times 10 \times 10\) product of only three numbers eg5 840 no products
B2 B1 B0 B0 B0
Products may be seen in any order eg1 \(2 \times 3 \times 8 \times 10\) eg2 \(10 \times 10 \times 2 \times 6\)
18 How many odd numbers greater than 30 000 can be formed from these digits
\[2 \qquad 4 \qquad 6 \qquad 7 \qquad 8\]
with no repetition of any digit? [3 marks]
Mark scheme
Answer
Mark
Comments
18
B3
B2 identifies there are 3 choices for first digit and 3 choices for second digit B1 identifies there are 3 choices for first digit or identifies there is 1 choice for last digit
