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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 10 - 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 10 - 2020 - Paper 1

Step 1

Selection of 6 Contestants

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Answer

First, we need to select 6 contestants out of the total 10. This can be done using the combination formula:

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

where n is the total number of contestants and r is the number of contestants to select. Here, it is:

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

Step 2

Arrangement of 4 Contestants

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Answer

Next, we need to arrange the selected 4 contestants in the first, second, third, and fourth places. The number of arrangements is given by the permutations formula:

P(n)=n!P(n) = n!

In our case, we will arrange 4 contestants:

P(4)=4!P(4) = 4!

Step 3

Total Combinations and Arrangements

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Finally, we multiply the number of combinations (selection) by the number of arrangements:

Total Ways=C(10,6)×P(4)=10!6!4!×4!=10!/6!\text{Total Ways} = C(10, 6) \times P(4) = \frac{10!}{6!4!} \times 4! = 10! / 6!

Thus, the answer is option A: 10!6!\frac{10!}{6!}.

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