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Ray has nine cards numbered 1 to 9 - Edexcel - GCSE Maths - Question 22 - 2022 - Paper 3
Question 22
Ray has nine cards numbered 1 to 9. Ray takes at random three of these cards. He works out the sum of the numbers on the three cards and records the result. Work ... show full transcript
Worked Solution & Example Answer:Ray has nine cards numbered 1 to 9 - Edexcel - GCSE Maths - Question 22 - 2022 - Paper 3
Step 1
Determine possible combinations of cards
Answer
To find the probability of obtaining an even sum through randomly selecting three cards from the nine available, we first need to establish how many total combinations of three cards can be drawn from nine. This can be calculated using the combination formula:
inom{n}{r} = \frac{n!}{r!(n - r)!}In this case, n = 9 (total cards) and r = 3 (cards to choose):
inom{9}{3} = \frac{9!}{3!(9 - 3)!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84Step 2
Identify conditions for an even sum
Answer
Next, we need to find how many combinations yield an even sum. An even sum can occur under two scenarios:
- All three selected numbers are even.
- One selected number is odd, and two selected numbers are even.
The even numbers from cards 1 to 9 are 2, 4, 6, and 8. The odd numbers are 1, 3, 5, 7, and 9.
Step 3
Calculate combinations leading to an even sum
Answer
- Three even numbers: The available even numbers are 2, 4, 6, and 8 (total 4 even numbers). The number of ways to choose three from these four:
- Two even and one odd: We have 4 even numbers and 5 odd numbers. Choosing 2 even from 4 and 1 odd from 5:
Adding these two cases together gives:
Total combinations yielding an even sum: 4 + 30 = 34.
Step 4