Question

There are 6 athletes on a cross country team. At a photoshoot, 2 of the athletes need to be chosen and arranged in the front row. In how many ways can this be done?

Answer 1

There are 30 different ways to choose and arrange 2 athletes from a team of 6 for a photoshoot, calculated by using the permutations formula.

Step-by-step explanation:

To determine the number of ways 2 athletes out of 6 on a cross country team can be chosen and arranged in the front row for a photoshoot, we can use the concept of permutations since the order of the athletes matters. The formula to calculate the number of permutations of r objects taken from a set of n objects is nPr = n! / (n-r)!.

For this scenario where n=6 (total athletes) and r=2 (athletes to be chosen and arranged), the permutation can be calculated as:

1. Calculate the factorial of n: 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720
2. Calculate the factorial of (n-r): (6-2)! = 4! = 4 × 3 × 2 × 1 = 24
3. Apply the permutation formula: 6P2 = 6! / (6-2)! = 720 / 24 = 30

Therefore, there are 30 different ways to arrange 2 athletes from a team of 6 in the front row.

Answer 2

Given:

Total athletes = 6

Chosen and arranged in row = 2

Find-:How many ways can this be done

Sol:

Chosen 2 athletes.

Combination without repetition is:

`=((n+r-1)!)/(r!(n-1)!)`

Where ,

`\begin{gathered} n=6 \\ \\ r=2 \end{gathered}`

So,

`\begin{gathered} =((6+2-1)!)/(2!(6-1)!) \\ \\ =(7!)/(2!*5!) \\ \\ =(7*6*5!)/(2*1*5!) \\ \\ =7*3 \\ \\ =21 \end{gathered}`

So total 21 ways