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Out of 10 contestants, six are to be selected for the final round of a competition - HSC - SSCE Mathematics Extension 1 - Question 8 - 2020 - Paper 1

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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 th... 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 8 - 2020 - Paper 1

Step 1

Calculate the number of ways to select contestants

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Answer

To select 6 out of 10 contestants, we can use the combination formula:

C(n,r)=n!r!(n−r)!C(n, r) = \frac{n!}{r!(n-r)!}

Here, (n = 10) and (r = 6):

C(10,6)=10!6!4!C(10, 6) = \frac{10!}{6!4!}

Step 2

Calculate the arrangements for 1st, 2nd, 3rd, and 4th places

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Answer

Once 6 contestants are selected, we need to arrange 4 of them in a specific order. This is a permutation problem:

The number of arrangements is given by (P(6, 4) = \frac{6!}{(6-4)!} = \frac{6!}{2!}$$.

Step 3

Total ways to select and arrange contestants

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Answer

Combining both results:

Total ways = (C(10, 6) \times P(6, 4) = \frac{10!}{6!4!} \times \frac{6!}{2!} = \frac{10!}{4!2!}$$.

Thus, the final answer corresponds to Option C: (\frac{10!}{4!2!}).

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