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Math  /  Discrete

QuestionHow many groups of 3 can you make out of 6 ?

Studdy Solution

STEP 1

1. We are dealing with combinations, not permutations, because the order of selection does not matter.
2. We use the combination formula to find the number of ways to choose 3 items from 6.

STEP 2

1. Understand the combination formula.
2. Apply the combination formula to the problem.

STEP 3

The combination formula is given by:
C(n,r)=n!r!(n−r)! C(n, r) = \frac{n!}{r!(n-r)!}
where n n is the total number of items to choose from, and r r is the number of items to choose.

STEP 4

In this problem, n=6 n = 6 and r=3 r = 3 . Substitute these values into the combination formula:
C(6,3)=6!3!(6−3)! C(6, 3) = \frac{6!}{3!(6-3)!}

STEP 5

Calculate the factorials:
6!=6×5×4×3×2×1=720 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 3!=3×2×1=6 3! = 3 \times 2 \times 1 = 6 (6−3)!=3!=3×2×1=6 (6-3)! = 3! = 3 \times 2 \times 1 = 6

STEP 6

Substitute the factorials back into the formula:
C(6,3)=7206×6=72036 C(6, 3) = \frac{720}{6 \times 6} = \frac{720}{36}

STEP 7

Perform the division:
72036=20 \frac{720}{36} = 20
The number of groups of 3 that can be made out of 6 is:
20 \boxed{20}

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