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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

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 from 10

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Answer

To select 6 contestants from 10, we use the combination formula given by:

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

Here, n = 10 and r = 6:

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

Step 2

Arrange the selected 4 contestants

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Answer

Once the 6 contestants are selected, we need to arrange 4 of them in order to assign the 1st, 2nd, 3rd, and 4th places. This is a permutation of 4 objects from 6, calculated as:

$P(6, 4) = \frac{6!}{(6 - 4)!} = \frac{6!}{2!}$

Step 3

Calculate the total arrangements

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Answer

The total number of ways to carry out the selection and arrangement is the product of the combination and permutation calculated earlier:

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

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