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Math  /  Numbers & Operations

QuestionEvaluate. 3!5!6!4!\frac{3!5!}{6!4!}
Simplify your answer as much as possible. □\square

Studdy Solution

STEP 1

1. We need to evaluate the expression 3!5!6!4!\frac{3!5!}{6!4!}.
2. Factorials are defined as n!=n×(n−1)×⋯×1n! = n \times (n-1) \times \cdots \times 1.
3. Simplification involves canceling common factors in the numerator and denominator.

STEP 2

1. Expand the factorials.
2. Simplify the expression by canceling common factors.
3. Calculate the simplified expression.

STEP 3

Expand each factorial in the expression:
3!=3×2×13! = 3 \times 2 \times 1 5!=5×4×3×2×15! = 5 \times 4 \times 3 \times 2 \times 1 6!=6×5×4×3×2×16! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 4!=4×3×2×14! = 4 \times 3 \times 2 \times 1

STEP 4

Substitute the expanded factorials into the expression:
3!5!6!4!=(3×2×1)(5×4×3×2×1)(6×5×4×3×2×1)(4×3×2×1)\frac{3!5!}{6!4!} = \frac{(3 \times 2 \times 1)(5 \times 4 \times 3 \times 2 \times 1)}{(6 \times 5 \times 4 \times 3 \times 2 \times 1)(4 \times 3 \times 2 \times 1)}

STEP 5

Cancel common factors in the numerator and denominator:
- 5×4×3×2×15 \times 4 \times 3 \times 2 \times 1 in the numerator cancels with part of 6×5×4×3×2×16 \times 5 \times 4 \times 3 \times 2 \times 1 in the denominator. - 4×3×2×14 \times 3 \times 2 \times 1 in the denominator cancels completely.
This simplifies to:
36\frac{3}{6}

STEP 6

Simplify 36\frac{3}{6} by dividing both the numerator and the denominator by their greatest common divisor, which is 3:
36=12\frac{3}{6} = \frac{1}{2}
The simplified value of the expression is:
12\boxed{\frac{1}{2}}

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