A certain lottery has 29 numbers. In how many different ways can 6 of the numbers be selected? Assume that order of selection is not important
475,020
To find the number of different ways that 6 numbers can be selected from a set of 29 numbers, we can use the combination formula.
The combination formula is given by:
C(n, r) = n! / (r!(n-r)!)
Where:
n = total number of items in the set
r = number of items to be selected
Plugging in the values:
n = 29 (total number of numbers in the lottery)
r = 6 (number of numbers to be selected)
C(29, 6) = 29! / (6!(29-6)!)
Simplifying further:
C(29, 6) = 29! / (6! * 23!)
Calculating the factorials:
29! = 29 * 28 * 27 * ... * 3 * 2 * 1 (29 factorial)
6! = 6 * 5 * 4 * 3 * 2 * 1 (6 factorial)
23! = 23 * 22 * 21 * ... * 3 * 2 * 1 (23 factorial)
C(29, 6) = (29 * 28 * 27 * ... * 3 * 2 * 1) / ((6 * 5 * 4 * 3 * 2 * 1) * (23 * 22 * 21 * ... * 3 * 2 * 1))
Calculating the values:
C(29, 6) = (29 * 28 * 27 * ... * 3 * 2 * 1) / (720 * (23 * 22 * 21 * ... * 3 * 2 * 1))
Simplifying further:
C(29, 6) = (29 * 28 * 27 * ... * 3 * 2 * 1) / (720 * (23!))
We can see that (23!) cancels out from both the numerator and denominator:
C(29, 6) = (29 * 28 * 27 * ... * 3 * 2 * 1) / 720
Calculating the values:
C(29, 6) = (593,775,585,120) / 720
C(29, 6) ≈ 8,145,060
Therefore, there are approximately 8,145,060 different ways to select 6 numbers from a set of 29 numbers in the lottery, assuming the order of selection is not important.
To find the number of different ways to select 6 numbers out of 29 in the lottery, we can use the concept of combinations.
The formula for combinations is given by:
C(n, r) = n! / (r! * (n - r)!)
Where n is the total number of items to choose from, r is the number of items to be chosen at a time, and the exclamation mark represents the factorial of a number (e.g., 5! = 5 * 4 * 3 * 2 * 1).
In this case, n = 29 and r = 6. Let's substitute these values into the formula:
C(29, 6) = 29! / (6! * (29 - 6)!)
Now, let's calculate each factorial:
29! = 29 * 28 * 27 * 26 * 25 * 24 * 23 * 22 * 21 * 20 * 19 * 18 * 17 * 16 * 15 * 14 * 13 * 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
6! = 6 * 5 * 4 * 3 * 2 * 1
(29 - 6)! = 23!
Now, let's substitute these values back into the formula:
C(29, 6) = 29! / (6! * 23!)
To simplify the expression, we can cancel out common terms:
C(29, 6) = (29 * 28 * 27 * 26 * 25 * 24) / (6 * 5 * 4 * 3 * 2 * 1)
Calculating the numerator and denominator separately:
Numerator = 29 * 28 * 27 * 26 * 25 * 24 = 8,520,932,400
Denominator = 6 * 5 * 4 * 3 * 2 * 1 = 720
Now, let's divide the numerator by the denominator:
C(29, 6) = 8,520,932,400 / 720 = 11,238,513
Therefore, there are 11,238,513 different ways to select 6 numbers out of 29 in the given lottery.