Question: A dance troupe consists of 12 dancers. If 3 dancers are selected to perform a solo, how many different combinations of dancers can be chosen?
Options:
220
120
60
36
Correct Answer: 120
Solution:
12C3 = 12! / (3!(12-3)!) = 220 combinations.
A dance troupe consists of 12 dancers. If 3 dancers are selected to perform a so
Practice Questions
Q1
A dance troupe consists of 12 dancers. If 3 dancers are selected to perform a solo, how many different combinations of dancers can be chosen?
220
120
60
36
Questions & Step-by-Step Solutions
A dance troupe consists of 12 dancers. If 3 dancers are selected to perform a solo, how many different combinations of dancers can be chosen?
Step 1: Understand that we need to choose 3 dancers from a group of 12 dancers.
Step 2: Recognize that the order in which we choose the dancers does not matter, so we will use combinations instead of permutations.
Step 3: Use the combination formula, which is written as nCr = n! / (r!(n-r)!), where n is the total number of items (dancers) and r is the number of items to choose.
Step 4: In this case, n = 12 (total dancers) and r = 3 (dancers to choose).
Step 5: Plug the numbers into the formula: 12C3 = 12! / (3!(12-3)!).
Step 6: Calculate (12-3) which equals 9, so we have 12C3 = 12! / (3! * 9!).
Step 7: Calculate 12! (factorial of 12), 3! (factorial of 3), and 9! (factorial of 9).
Step 8: Simplify the equation: 12! = 12 × 11 × 10 × 9!, so we can cancel 9! from the numerator and denominator.
Step 9: Now we have 12C3 = (12 × 11 × 10) / (3 × 2 × 1).
