Question

evaluate.
\\(\frac{6!10!}{8!8!}\\)
simplify your answer as much as possible.
your answer

Explanation:

Step1: Recall factorial definition

A factorial \( n! = n\times(n - 1)\times\cdots\times1 \). So we can rewrite \( 10! = 10\times9\times8! \) and \( 8! = 8\times7\times6! \) (wait, actually for simplification, let's handle numerator and denominator:

Numerator: \( 6! \times 10! = 6! \times 10\times9\times8! \)

Denominator: \( 8! \times 8! \)

So substitute these into the fraction:

\( \frac{6! \times 10\times9\times8!}{8! \times 8!} \)

Now, cancel out one \( 8! \) from numerator and denominator:

\( \frac{6! \times 10\times9}{8!} \)

But \( 8! = 8\times7\times6! \), so substitute that in:

\( \frac{6! \times 10\times9}{8\times7\times6!} \)

Now cancel out \( 6! \) from numerator and denominator:

\( \frac{10\times9}{8\times7} \)

Step2: Calculate the numerator and denominator

Numerator: \( 10\times9 = 90 \)

Denominator: \( 8\times7 = 56 \)

Now simplify the fraction \( \frac{90}{56} \). Both are divisible by 2:

\( \frac{90\div2}{56\div2} = \frac{45}{28} \)

Wait, wait, did I make a mistake? Wait, let's re-examine the original expression: \( \frac{6!10!}{8!8!} \)

Alternative approach:

\( 10! = 10\times9\times8\times7\times6! \)? No, \( 10! = 10\times9\times8! \), because \( 8! = 8\times7\times6\times5\times4\times3\times2\times1 \), so \( 10! = 10\times9\times8\times7\times6\times5\times4\times3\times2\times1 = 10\times9\times8! \)

Similarly, \( 8! = 8\times7\times6! \), because \( 6! = 6\times5\times\cdots\times1 \), so \( 8! = 8\times7\times6! \)

So let's rewrite the numerator and denominator with these:

Numerator: \( 6! \times 10! = 6! \times 10\times9\times8! \)

Denominator: \( 8! \times 8! = 8! \times 8\times7\times6! \)

Now substitute into the fraction:

\( \frac{6! \times 10\times9\times8!}{8! \times 8\times7\times6!} \)

Now, cancel \( 6! \) from numerator and denominator, cancel one \( 8! \) from numerator and denominator:

We get \( \frac{10\times9}{8\times7} \)

Calculate that: \( 10\times9 = 90 \), \( 8\times7 = 56 \), so \( \frac{90}{56} = \frac{45}{28} \). Wait, but let's check with actual factorial values:

\( 6! = 720 \), \( 10! = 3628800 \), \( 8! = 40320 \)

So numerator: \( 720 \times 3628800 = 720 \times 3628800 \)

Denominator: \( 40320 \times 40320 \)

Calculate numerator: \( 720 \times 3628800 = 720 \times 3.6288\times10^6 = 2.612736\times10^9 \)

Denominator: \( 40320 \times 40320 = (4.032\times10^4)^2 = 1.6257024\times10^9 \)

Now divide numerator by denominator: \( \frac{2.612736\times10^9}{1.6257024\times10^9} \approx 1.607 \), but \( \frac{45}{28} \approx 1.607 \), so that's correct. Wait, but let's do the fraction again:

Wait, \( 10! = 10\times9\times8! \), \( 8! = 8\times7\times6! \)

So:

\( \frac{6! \times 10\times9\times8!}{8! \times 8\times7\times6!} = \frac{10\times9}{8\times7} = \frac{90}{56} = \frac{45}{28} \)

Yes, that's correct. So the simplified fraction is \( \frac{45}{28} \) or as a mixed number \( 1\frac{17}{28} \), but as an improper fraction, \( \frac{45}{28} \) is simplified.

Wait, but let's check the steps again. Maybe I messed up the factorial expansion. Let's do it step by step:

\( 6! = 720 \)

\( 10! = 3628800 \)

\( 8! = 40320 \)

So numerator: \( 720 \times 3628800 = 720 \times 3628800 = 2612736000 \)

Denominator: \( 40320 \times 40320 = 1625702400 \)

Now divide 2612736000 by 1625702400:

Divide numerator and denominator by 1000: 2612736 / 1625702.4 → no, better to divide numerator and denominator by 4800 (wait, maybe GCD of 2612736000 and 1625702400). Let's find GCD of 2612736000 and 1625702400.

Divide 2612736000 by 1625702400: quotient 1, remainder 2612736000 - 1625702400 = 987033600

Now GCD(1625702400, 987033600)

1625702400 ÷ 987033600 = 1, remainder 1625702400 - 987033600 = 638668800

GCD(987033600, 638668800)

987033600 ÷ 638668800 = 1, remainder 348364800

GCD(638668800, 348364800)

638668800 ÷ 348364800 = 1, remainder 290304000

GCD(348364800, 290304000)

348364800 ÷ 290304000 = 1, remainder 58060800

GCD(290304000, 58060800)

290304000 ÷ 58060800 = 5, remainder 290304000 - 5×58060800 = 290304000 - 290304000 = 0

So GCD is 58060800

Now divide numerator and denominator by 58060800:

Numerator: 2612736000 ÷ 58060800 = 45

Denominator: 1625702400 ÷ 58060800 = 28

Ah, so that confirms it: 45/28. So the simplified fraction is 45/28.

Answer:

\(\frac{45}{28}\)