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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 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

Selecting 6 contestants from 10

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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 $n$ is the total number of contestants (10) and $r$ is the number selected (6).

Thus, the number of ways to select 6 contestants is:

inom{10}{6} = rac{10!}{6!4!} = 210

Step 2

Arranging 4 placement spots

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Answer

Once the 6 contestants are selected, we need to arrange 4 of them into specific positions (1st, 2nd, 3rd, and 4th).

The number of ways to arrange 4 contestants from 6 is given by the permutation formula:

P(n, r) = rac{n!}{(n - r)!}

In this case, it is:

P(6, 4) = rac{6!}{(6 - 4)!} = rac{6!}{2!} = 360

Step 3

Total number of arrangements

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Answer

To find the total number of ways to carry out this process, we multiply the number of selections by the number of arrangements:

Total = inom{10}{6} imes P(6, 4) = 210 imes 360 = 75600

Thus, the answer corresponds to option A.

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