if one can choose any six of 40 numbers, in any order, to play a state lottery game, how many different arrangments are possible?
To determine the number of different arrangements possible when choosing any six numbers from a set of 40, we can use combination and permutation principles.
First, let's understand the difference between combinations and permutations:
- Combinations: This refers to the number of ways to select items from a larger set without considering the order. In other words, the order of the items selected does not matter.
- Permutations: This refers to the number of ways to arrange items from a larger set, considering the order of the items.
In this case, we need to calculate the number of combinations because the order of the numbers selected for the lottery game does not matter.
The formula for combinations is given by:
C(n, k) = n! / (k! * (n-k)!)
Where:
- n is the total number of items in the set
- k is the number of items selected
In this case, we have n = 40 (40 numbers to choose from) and k = 6 (selecting any 6 numbers).
Using the formula, we can calculate the number of combinations as follows:
C(40, 6) = 40! / (6! * (40-6)!)
Simplifying the equation:
C(40, 6) = (40 * 39 * 38 * 37 * 36 * 35) / (6 * 5 * 4 * 3 * 2 * 1)
Now we can calculate:
C(40, 6) = 3,838,380
Therefore, there are 3,838,380 different arrangements possible when choosing any six numbers in any order from a set of 40 numbers for the state lottery game.