4. An environmental agency will randomly select 4 houses from a block containing 25 houses for a radon check.
a) How many total selections are possible?
b) How many permutations are possible?
a. C(25,4) = P(25,4)/4!
b. P(25,4) = 25*24*23*22
a
a) To calculate the total number of selections, we need to use the concept of combinations. The formula for combinations is:
nCr = n! / (r!(n-r)!)
Where n is the total number of houses (25 in this case) and r is the number of houses to be selected (4 in this case).
Plugging in the values, we get:
25C4 = 25! / (4!(25-4)!)
= 25! / (4!*21!)
Calculating this, we find that there are 12,650 total selections possible.
b) To calculate the number of permutations, we use the formula:
nPn = n!
Where n is the total number of houses (25 in this case) and Pn is the number of permutations possible for selecting all houses from the block.
Plugging in the values, we get:
25P25 = 25!
Calculating this, we find that there are 15,511,210,043,330,985,984,000,000 possible permutations.
To find the total number of selections and permutations in this scenario, we need to use the concepts of combinations and permutations.
a) To calculate the total number of selections, we need to find the number of possible combinations. This is because the order in which the houses are selected does not matter. We can use the combination formula for this:
C(n, r) = n! / (r!(n-r)!)
In this case, there are 25 houses (n) to choose from, and we want to select 4 houses (r). Plugging these values into the formula:
C(25, 4) = 25! / (4!(25-4)!) = 25! / (4!21!)
Simplifying this expression:
C(25, 4) = 25 * 24 * 23 * 22 / (4 * 3 * 2 * 1) = 12,650
Therefore, there are 12,650 possible selections of 4 houses from a block containing 25 houses.
b) To calculate the number of permutations, we need to consider the order in which the houses are selected. We can use the permutation formula for this:
P(n, r) = n! / (n-r)!
In this case, there are 25 houses (n) to choose from, and we want to select 4 houses (r). Plugging these values into the formula:
P(25, 4) = 25! / (25-4)!
Simplifying this expression:
P(25, 4) = 25 * 24 * 23 * 22 = 303,600
Therefore, there are 303,600 possible permutations of selecting 4 houses from a block containing 25 houses.