Photo AI

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 icon

Question 10

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.png

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

96%

114 rated

Answer

The first step is to select 6 contestants from the 10 available. This can be calculated using the combination formula:

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

For our situation, it will be:

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

Step 2

Arrange the selected contestants

99%

104 rated

Answer

Once we have selected the 6 contestants, we need to arrange 4 of them in 1st, 2nd, 3rd, and 4th places. This is a permutation calculation and can be expressed as:

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

Thus, we need:

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

Step 3

Calculate the total ways

96%

101 rated

Answer

Now, to find the total number of ways to carry out this process, we multiply the results of the two steps above:

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

Substituting our previous calculations yields:

Total=10!6!4!×6!2!Total = \frac{10!}{6!4!} \times \frac{6!}{2!}

This simplifies to:

Total=10!4!2!Total = \frac{10!}{4!2!}

This gives us the total arrangements for the selected contestants.

Join the SSCE students using SimpleStudy...

97% of Students

Report Improved Results

98% of Students

Recommend to friends

100,000+

Students Supported

1 Million+

Questions answered

;