Combinations: Quick Revision

A brief note on Combinations. Read it before a test. Try each question at the end before you open its answer.

The formula

• A combination is a selection: order does not matter.
• $^{n}C_{r} = \dfrac{n!}{r!\,(n - r)!} = \dfrac{n(n - 1)\cdots(n - r + 1)}{r!}$.
• $^{n}P_{r} = {}^{n}C_{r} \times r!$: choose, then arrange.
• $^{n}C_{0} = {}^{n}C_{n} = 1$, $^{n}C_{1} = n$ and $^{n}C_{2} = \dfrac{n(n - 1)}{2}$.

Properties

Geometry

• $n$ points, no three in a line: $^{n}C_{2}$ lines and $^{n}C_{3}$ triangles.
• A polygon of $n$ sides: $^{n}C_{2} - n = \dfrac{n(n - 3)}{2}$ diagonals.
• If $k$ of the points lie on one line: subtract $^{k}C_{3}$ triangles, and replace $^{k}C_{2}$ lines by $1$.

Selections with conditions

• Must be in: put them in, choose the rest from the others.
• Must be out: choose everything from the others.
• Separate groups: multiply the counts.
• At least one: total minus none.
• Words and codes: select, then arrange.

A worked example

A quiz team of $4$ is chosen from $9$ girls and $5$ boys. How many teams have at least one boy?

Step 1. All teams. $^{14}C_{4} = 1001$.

Step 2. No boy. $^{9}C_{4} = 126$.

Step 3. Subtract. $1001 - 126 = 875$.

Practice with new numbers

A program to try

The program lists every quiz team of the worked example, keeps those with a boy, and checks the count against the formula.

Mistakes to avoid

• $^{n}P_{r}$ for a team. A team has no order. Use $^{n}C_{r}$.
• Only $a = b$. $^{n}C_{a} = {}^{n}C_{b}$ also allows $a + b = n$.
• Sides as diagonals. Take the $n$ sides away from $^{n}C_{2}$.
• One chosen first. Picking one woman first counts some committees more than once.
• Selecting without arranging. Words need the $r!$ as well.

Check yourself

1. Work out $^{17}C_{15}$.

2. If $^{n}C_{2} = 28$, what is $n$?

3. How many diagonals does a polygon with $9$ sides have?

4. Choose $2$ of $6$ singers and $1$ of $4$ drummers for a band. In how many ways?

5. Write $^{7}C_{2} + {}^{7}C_{3}$ as one term, and find its value.