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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 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

Step 1: Selecting the Contestants

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Answer

First, we need to choose 6 contestants from a total of 10. The number of ways to choose 6 from 10 can be calculated using the combination formula, which is given by:

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

Here, we have:

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

Step 2

Step 2: Arranging the Selected Contestants

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Answer

After selecting the contestants, we need to arrange 4 of them in the 1st, 2nd, 3rd, and 4th positions. The number of ways to arrange 4 contestants is given by the factorial of 4, which is:

4!4!

Step 3

Step 3: Total Number of Ways

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Answer

The total number of ways to select and arrange the contestants is the product of the results from Step 1 and Step 2:

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

Step 4

Final Conclusion

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Answer

Thus, the total number of ways this process can be carried out is:

10!6!\frac{10!}{6!}

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