The number of students taking German or Spanish is 9 + 9 = 18. Of that group of 18, 6 are taking both languages so they've been counted twice (once in the German group and once in the Spanish group). Subtracting them out leaves 18 - 6 = 12 who are taking at least one language. 22 - 12 = 10 students who are not taking either language.

To fill a 10 gallon tank exactly halfway you'll need 5 gallons of fuel. Each fuel can holds 2\(\frac{1}{2}\) gallons so:

cans = \( \frac{5 \text{ gallons}}{2\frac{1}{2} \text{ gallons}} \) = 2

The greatest common factor (GCF) is the greatest factor that divides two integers.

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:

\( \frac{2!}{4!} \)
\( \frac{2 \times 1}{4 \times 3 \times 2 \times 1} \)
\( \frac{1}{4 \times 3} \)
\( \frac{1}{12} \)

To add 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 9 are [9, 18, 27, 36, 45, 54, 63, 72, 81, 90]. The first few multiples they share are [9, 18, 27, 36, 45] making 9 the smallest multiple 3 and 9 share.

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

\( \frac{8 x 3}{3 x 3} \) + \( \frac{2 x 1}{9 x 1} \)

\( \frac{24}{9} \) + \( \frac{2}{9} \)

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

\( \frac{24 + 2}{9} \) = \( \frac{26}{9} \) = 2\(\frac{8}{9}\)