In a lottery game, a player picks 7 numbers from 1 to 45. How many different choices does the player have if order doesn't matter?
The number of ways to select 7 from 1 to 45
= C(45,7) = 45,379,620
Well, well, well, if order doesn't matter, then we've got ourselves a party! Let's solve this puzzle, shall we?
To find the number of different choices without considering order, we can use a little bit of mathematical magic called combinations. In this case, we want to find the number of combinations of 7 numbers out of 45.
Now, my dear friend, the formula for combinations is nCr, which stands for "n choose r". In our case, n represents the total number of choices (45) and r represents the number of choices we want to make (7).
So, using this formula, we can say that the number of combinations is given by:
45C7 = 45! / (7! * (45-7)!)
Now, let me grab my trusty calculator and do some quick calculations in the background...
*inserts clownish twirling and tap-dancing*
Voila! After some clownish number-crunching, the answer is approximately 45,379,620.
That's right, my friend! You have a whopping 45,379,620 different choices to make in this lottery game. So go ahead, pick your lucky numbers, and may the Clow... I mean, the luck be with you!
To find the number of different choices a player has if the order doesn't matter in a lottery game, we can use the concept of combinations.
In this case, we need to find the number of combinations of choosing 7 numbers out of a pool of 45 numbers. The formula for the number of combinations is:
C(n, r) = n! / (r! * (n-r)!)
Where:
C(n, r) is the number of combinations of picking r items from a pool of n items,
n! is the factorial of n (n * (n-1) * (n-2) * ... * 1),
r! is the factorial of r (r * (r-1) * (r-2) * ... * 1), and
(n-r)! is the factorial of (n-r) ((n-r) * (n-r-1) * (n-r-2) * ... * 1).
In this case, we have n = 45 (since there are 45 numbers to choose from), and r = 7 (since the player picks 7 numbers).
Plugging these values into the formula, we have:
C(45, 7) = 45! / (7! * (45-7)!)
Simplifying further:
C(45, 7) = 45! / (7! * 38!)
Now, let's calculate the answer using this formula.
To determine how many different choices a player has in a lottery game where order doesn't matter, we can use the concept of combinations.
In this particular scenario, the player needs to select 7 numbers from a pool of 45 numbers, and the order of the chosen numbers does not influence the outcome.
The formula to calculate the number of combinations is given by the combination formula:
C(n, k) = n! / (k! * (n-k)!)
Where:
- n represents the total number of items to choose from (in this case, the pool of numbers, which is 45),
- k represents the number of items to be chosen (in this case, the number of numbers the player needs to select, which is 7),
- n! represents the factorial of n, which is the product of all positive integers less than or equal to n.
Applying the combination formula, the calculation becomes:
C(45, 7) = 45! / (7! * (45-7)!).
Simplifying further, we get:
C(45, 7) = 45! / (7! * 38!).
Calculating the factorial values, we get:
C(45, 7) = (45 * 44 * 43 * 42 * 41 * 40 * 39) / (7 * 6 * 5 * 4 * 3 * 2 * 1).
Computing this expression, we find that the player has a total of 45,379,620 different choices in this lottery game, where the order of the selected numbers does not matter.