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Question Evaluate the factorial expression 2! 5!6! 3!\frac{2! \, 5!}{6! \, 3!} and simplify the result.

Studdy Solution

STEP 1

1. The factorial of a non-negative integer nn, denoted as n!n!, is the product of all positive integers less than or equal to nn.
2. Factorials can be simplified by dividing out common factors.
3. The expression 2!5!6!3!\frac{2!5!}{6!3!} can be simplified by recognizing that 6!6! and 5!5! share common factors, as do 3!3! and 2!2!.

STEP 2

1. Recognize and cancel out common factors in the numerator and denominator.
2. Simplify the resulting expression.

STEP 3

Write out the factorials explicitly to identify common factors.
2!5!6!3!=(1⋅2)(1⋅2⋅3⋅4⋅5)(1⋅2⋅3⋅4⋅5⋅6)(1⋅2⋅3) \frac{2!5!}{6!3!} = \frac{(1 \cdot 2)(1 \cdot 2 \cdot 3 \cdot 4 \cdot 5)}{(1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6)(1 \cdot 2 \cdot 3)}

STEP 4

Cancel out the common factors in the numerator and denominator.
(1⋅2)(1⋅2⋅3⋅4⋅5)(1⋅2⋅3⋅4⋅5⋅6)(1⋅2⋅3)=2⋅5!6⋅5!⋅3!=16⋅3! \frac{(1 \cdot 2)(1 \cdot 2 \cdot 3 \cdot 4 \cdot 5)}{(1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6)(1 \cdot 2 \cdot 3)} = \frac{2 \cdot 5!}{6 \cdot 5! \cdot 3!} = \frac{1}{6 \cdot 3!}

STEP 5

Simplify the remaining factorial in the denominator.
16⋅3!=16⋅(1⋅2⋅3)=16⋅6 \frac{1}{6 \cdot 3!} = \frac{1}{6 \cdot (1 \cdot 2 \cdot 3)} = \frac{1}{6 \cdot 6}

STEP 6

Simplify the final expression.
16⋅6=136 \frac{1}{6 \cdot 6} = \frac{1}{36}
The simplified expression is 136\frac{1}{36}.

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