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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 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 8 - 2020 - Paper 1

Step 1

How many ways to choose 6 contestants

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Answer

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

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

Where:

  • n = total contestants = 10
  • k = contestants to select = 6

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

Step 2

How to arrange 4 out of the selected 6

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Answer

Next, to arrange 4 contestants in 1st, 2nd, 3rd, and 4th positions, we use the permutation formula:

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

Where:

  • n = contestants chosen = 6
  • r = positions to fill = 4

Calculating: P(6,4)=6!(6−4)!=6!/2!P(6, 4) = \frac{6!}{(6-4)!} = 6! / 2!

Step 3

Final calculation

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Answer

To find the total number of ways to select and arrange, we multiply the two results:

Total Ways=C(10,6)×P(6,4)Total\ Ways = C(10, 6) \times P(6, 4)

Substituting the values: Total Ways=10!6!4!×6!2!Total\ Ways = \frac{10!}{6!4!} \times \frac{6!}{2!} After simplifying, we will arrive at the final answer.

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