To win one version of the New York State lottery, one must correctly select 5 numbers from 49 numbers. How many different selections are possible if:
~ The order in which the selection is made does not matter?
~ The order of selection does matter?
When the order doesn't matter, it means we are dealing with combinations. When the order does matter, it means we are dealing with permutations.
1. When the order doesn't matter (combinations):
To find the number of combinations, we can use the formula for calculating combinations, which is nCr = n! / r!(n-r)!, where n is the total number of items and r is the number of items being chosen.
In this case, we need to select 5 numbers out of 49 without considering the order. So the formula becomes 49C5.
Calculating 49C5:
49! / (5!(49-5)!)
= 49! / (5!44!)
= (49 * 48 * 47 * 46 * 45) / (5 * 4 * 3 * 2 * 1)
= 1,906,884 different combinations.
Therefore, there are 1,906,884 different selections possible when the order doesn't matter.
2. When the order does matter (permutations):
To find the number of permutations, we can use the formula for calculating permutations, which is nPr = n! / (n-r)!, where n is the total number of items and r is the number of items being chosen.
In this case, we need to select 5 numbers out of 49 considering the order. So the formula becomes 49P5.
Calculating 49P5:
49! / (49-5)!
= 49! / 44!
= (49 * 48 * 47 * 46 * 45)
= 1,221,759,300 different permutations.
Therefore, there are 1,221,759,300 different selections possible when the order does matter.
The number of different selections possible in the New York State lottery depends on whether the order of selection matters or not.
1. If the order in which the selection is made does not matter:
To determine the number of ways to select 5 numbers out of 49 without regard to the order, we can use the combination formula.
The formula for combination is given by:
C(n, r) = n! / (r! * (n - r)!)
Where:
n = total number of items to choose from
r = number of items to select
In this case, since we are selecting 5 numbers out of 49, the formula becomes:
C(49, 5) = 49! / (5! * (49 - 5)!)
Simplifying this equation:
C(49, 5) = 49! / (5! * 44!)
Using factorials:
C(49, 5) = (49 * 48 * 47 * 46 * 45) / (5 * 4 * 3 * 2 * 1)
Cancelling out the common factors:
C(49, 5) = 49 * 48 * 47 * 46 * 45 / (5 * 4 * 3 * 2 * 1)
The total number of different selections possible in the New York State lottery, where the order does not matter, is approximately 1,906,884.
2. If the order of selection does matter:
To determine the number of ways to select 5 numbers out of 49 with regard to the order, we can use the permutation formula.
The formula for permutation is given by:
P(n, r) = n! / (n - r)!
Using this formula, we can calculate the number of different selections possible:
P(49, 5) = 49! / (49 - 5)!
Simplifying:
P(49, 5) = 49! / 44!
Using factorials:
P(49, 5) = (49 * 48 * 47 * 46 * 45) / (5 * 4 * 3 * 2 * 1)
Cancelling out the common factors:
P(49, 5) = 49 * 48 * 47 * 46 * 45 / (5 * 4 * 3 * 2 * 1)
The total number of different selections possible in the New York State lottery, where the order does matter, is approximately 2,394,044,880.