Out of 10 contestants, six are to be selected for the final round of a competition - HSC - SSCE Mathematics Extension 1 - Question 10 - 2020 - Paper 1
Question 10
Out of 10 contestants, six are to be selected for the final round of a competition. Four of those six will be placed 1st, 2nd, 3rd and 4th.
In how many ways can thi... show full transcript
Worked Solution & Example Answer:Out of 10 contestants, six are to be selected for the final round of a competition - HSC - SSCE Mathematics Extension 1 - Question 10 - 2020 - Paper 1
Step 1
Select 6 Contestants
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Answer
To select 6 contestants out of 10, we use the combination formula:
inom{n}{r} = rac{n!}{r!(n-r)!}
Where:
n=10 (total contestants)
r=6 (contestants to be selected)
Calculating:
inom{10}{6} = rac{10!}{6!4!} = 210
Step 2
Arrange 4 Contestants
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Answer
Once we have selected the 6 contestants, we need to arrange 4 of them in 1st, 2nd, 3rd, and 4th places. The number of ways to arrange 4 contestants is given by the permutation formula:
P(n, r) = rac{n!}{(n-r)!}
Where n=6 (selected contestants) and r=4.
Calculating:
P(6, 4) = rac{6!}{(6-4)!} = rac{6!}{2!} = 360
Step 3
Total Ways
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Answer
To find the total number of ways to select and arrange the contestants, we multiply the number of combinations by the number of permutations:
