A factorial is the product of an integer and all the positive integers below it. To solve a fraction featuring factorials, expand the factorials and cancel out like numbers:

\( \frac{2!}{6!} \)
\( \frac{2 \times 1}{6 \times 5 \times 4 \times 3 \times 2 \times 1} \)
\( \frac{1}{6 \times 5 \times 4 \times 3} \)
\( \frac{1}{360} \)

To multiply fractions, multiply the numerators together and then multiply the denominators together:

\( \frac{4}{9} \) x \( \frac{1}{6} \) = \( \frac{4 x 1}{9 x 6} \) = \( \frac{4}{54} \) = \(\frac{2}{27}\)

The hourly error rate for this machine is the error rate in parts per 100 multiplied by the number of parts produced per hour:

\( \frac{8}{100} \) x 9 = \( \frac{8 \times 9}{100} \) = \( \frac{72}{100} \) = 0.72 errors per hour

So, in an average hour, the machine will produce 9 - 0.72 = 8.28 error free parts.

The machine ran for 24 - 2 = 22 hours yesterday so you would expect that 22 x 8.28 = 182.2 error free parts were produced yesterday.

To subtract these fractions, first find the lowest common multiple of their denominators. The first few multiples of 3 are [3, 6, 9, 12, 15, 18, 21, 24, 27, 30] and the first few multiples of 11 are [11, 22, 33, 44, 55, 66, 77, 88, 99]. The first few multiples they share are [33, 66, 99] making 33 the smallest multiple 3 and 11 share.

Next, convert the fractions so each denominator equals the lowest common multiple:

\( \frac{7 x 11}{3 x 11} \) - \( \frac{7 x 3}{11 x 3} \)

\( \frac{77}{33} \) - \( \frac{21}{33} \)

Now, because the fractions share a common denominator, you can subtract them:

\( \frac{77 - 21}{33} \) = \( \frac{56}{33} \) = 1\(\frac{23}{33}\)

Average speed in miles per hour is the number of miles traveled divided by the number of hours:

Solving for distance:

distance = \( \text{speed} \times \text{time} \)
distance = \( 40mph \times 2h \)
80 miles