Evaluate the combination.
its in parentheses with the ten on top, and the 6 on the bottom, they are both in the same set of parentheses
(10)
(6 )
a. 5040 b. 210 c. 75,600 d. 48
To evaluate the combination, you need to calculate the value of the expression (10 choose 6). The expression "n choose r" represents the number of ways to choose r items from a set of n distinct items, where order does not matter and repetition is not allowed. To calculate this, you can use the formula:
n! / (r! * (n-r)!)
where "!" denotes the factorial function. The factorial of a number n is the product of all positive integers less than or equal to n.
In this case, you have (10 choose 6), so the calculation would be:
10! / (6! * (10-6)!)
Now let's break down the calculations step by step:
Step 1: Calculate the factorial of 10:
10! = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
Step 2: Calculate the factorial of 6:
6! = 6 * 5 * 4 * 3 * 2 * 1
Step 3: Calculate the factorial of (10-6):
(10-6)! = 4! = 4 * 3 * 2 * 1
Step 4: Substitute the values into the formula:
10! / (6! * (10-6)!) = (10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / ((6 * 5 * 4 * 3 * 2 * 1) * (4 * 3 * 2 * 1))
Step 5: Cancel out common factors:
The common factors cancel out, leaving:
= 10 * 9 * 8 * 7 / (4 * 3 * 2 * 1)
Step 6: Simplify the expression:
= 10 * 9 * 8 * 7 / 24
Step 7: Perform the multiplication and division:
= 5040 / 24
Step 8: Calculate the final result:
= 210
Therefore, the combination (10 choose 6) evaluates to 210. So, the correct answer is option b.