In how many ways can 5 people be chosen and arranged in a straight line, if there are 11 people from whom to choose?
First of all, the 5 people can be chosen from the 11 in
C(11,5) or 462 ways
so there are 462 groups of 5, but each of these can be arranged in 5! ways
so total number of ways for your situation is 462x5! = 55440
To calculate the number of ways to choose and arrange 5 people out of 11 in a straight line, we can use the concept of permutations.
The number of ways to choose 5 people out of 11 is denoted as "11 choose 5" and can be calculated using the combination formula:
nCr = n! / (r!(n - r)!)
Here, n represents the total number of options (11) and r represents the number of selections (5).
Plugging in the values, we get:
11C5 = 11! / (5!(11 - 5)!)
Simplifying further:
11C5 = 11! / (5! * 6!)
Now, let's calculate the factorials:
11! = 11 * 10 * 9 * 8 * 7 * 6!
5! = 5 * 4 * 3 * 2 * 1
Substituting the factorials back into the equation:
11C5 = (11 * 10 * 9 * 8 * 7 * 6!) / (5 * 4 * 3 * 2 * 1 * 6!)
A large portion of the numerators and denominators cancel out, simplifying the equation to:
11C5 = (11 * 10 * 9 * 8 * 7) / (5 * 4 * 3 * 2 * 1)
Calculating this further:
11C5 = 77,000 / 120
Simplifying the fraction:
11C5 = 641
Therefore, there are 641 ways to choose and arrange 5 people in a straight line out of a group of 11 people.
To find the number of ways to choose and arrange 5 people in a straight line from a pool of 11 people, we can use the concept of permutations.
The number of ways to choose and arrange 5 people can be calculated by using the formula for permutations:
P(n, r) = n! / (n - r)!
Where P(n, r) represents the number of permutations of choosing r objects from a pool of n objects.
In this case, we have a pool of 11 people (n) and we want to choose and arrange 5 people (r). So, we can calculate the number of ways as:
P(11, 5) = 11! / (11 - 5)!
Calculating further:
P(11, 5) = 11! / 6!
Now, let's break down the factorial expressions:
11! = 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
6! = 6 * 5 * 4 * 3 * 2 * 1
Substituting these values back into the formula:
P(11, 5) = (11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / (6 * 5 * 4 * 3 * 2 * 1)
Simplifying:
P(11, 5) = 11 * 10 * 9 * 8 * 7
Therefore, there are 11 * 10 * 9 * 8 * 7 = 55,440 ways to choose and arrange 5 people from a pool of 11 people in a straight line.