An executive committee of three is selected from a group of ten. Which expression best describes the number of combinations?
A. 10P3 (small 10 and 3)
B. 10!
C. 3P10 (small 3 and 10)
D. 10P3/3! (small 10 and 3 on top, big 3! on the bottom)
I am not sure what your notation is but here is the recipe:
C(n,r) = n!/[r!(n-r)!]
n = 10
r = 3
10!/[3!7!]
10*9*8/(3*2)
10*3*4
120
The correct expression that describes the number of combinations is option D: 10P3/3!.
The expression that best describes the number of combinations for selecting an executive committee of three from a group of ten is option D: 10P3/3!.
To understand why this is the correct expression, we first need to understand what each of the notations represents:
- "nCr" or "C(n,r)" represents the number of combinations of selecting r elements from a set of n elements. This can also be written as n! / (r! * (n-r)!).
- "nPr" or "P(n,r)" represents the number of permutations of selecting r elements from a set of n elements. This can be calculated as n! / (n-r)!
Now let's break down each of the options:
A. 10P3: This represents the number of permutations of selecting 3 elements from a set of 10. However, since we are looking for combinations (where the order of selection does not matter), this option is not correct.
B. 10!: This represents the factorial of 10, which is the product of all positive integers from 1 to 10. However, using this notation considers the order of selection, which is not applicable in our case. Therefore, this option is not correct.
C. 3P10: This represents the number of permutations of selecting 10 elements from a set of 3. Once again, since we are looking for combinations, this is not the correct expression.
D. 10P3/3!: This expression represents the number of permutations of selecting 3 elements from a set of 10, divided by the number of permutations of those same 3 elements. Dividing by 3! accounts for the fact that those 3 elements can be arranged in 3! = 6 different orders, but since we are only interested in the combinations, we want to eliminate the duplicates caused by different permutations. Hence, this expression correctly represents the number of combinations.
Therefore, the correct expression is option D: 10P3/3!.