Fractions and Decimals

Discrete Mathematics

Differential Equations

Solved on Mar 20, 2024

Evaluate and simplify the expression $\frac{7 ! 5 !}{4 ! 6 !}$ .

STEP 1

1. The notation $n!$ represents the factorial of a non-negative integer $n$ , which is the product of all positive integers less than or equal to $n$ .
2. Factorials can be simplified by canceling common factors in the numerator and denominator.

STEP 2

1. Expand the factorials in the numerator and denominator.
2. Cancel common factors.
3. Simplify the expression to its lowest terms.

STEP 3

Expand the factorials in the numerator and denominator.
$\frac{7! \cdot 5!}{4! \cdot 6!} = \frac{(7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1) \cdot (5 \cdot 4 \cdot 3 \cdot 2 \cdot 1)}{(4 \cdot 3 \cdot 2 \cdot 1) \cdot (6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1)}$

STEP 4

Cancel common factors in the numerator and denominator.
$\frac{(7 \cdot \cancel{6} \cdot \cancel{5} \cdot \cancel{4} \cdot \cancel{3} \cdot \cancel{2} \cdot \cancel{1}) \cdot (\cancel{5} \cdot \cancel{4} \cdot \cancel{3} \cdot \cancel{2} \cdot \cancel{1})}{(\cancel{4} \cdot \cancel{3} \cdot \cancel{2} \cdot \cancel{1}) \cdot (\cancel{6} \cdot \cancel{5} \cdot \cancel{4} \cdot \cancel{3} \cdot \cancel{2} \cdot \cancel{1})}$

STEP 5

Simplify the expression to its lowest terms.
$\frac{7! \cdot 5!}{4! \cdot 6!} = \frac{7}{1}$
The simplified expression is $7$ .

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