Combinations

Name: Combinations
Author: StemPy Lab

Selections when order does not matter: the formula for nCr, its properties, and problems that mix selecting and arranging.

3 lessons, about 1 h 15 min3 simulations12 problems3 practice questionsPython in every lesson

Course outline (3 lessons)

Lesson 1about 25 min
Combinations and the formula for nCr
A combination is a selection in which order does not matter. Each selection of $r$ things can be arranged in $r!$ orders, so $^{n}P_{r} = {}^{n}C_{r} \times r!$ and $^{n}C_{r} = \dfrac{n!}{r!\,(n - r)!}$ . In particular $^{n}C_{0} = {}^{n}C_{n} = 1$ and $^{n}C_{1} = n$ .
1 simulation1 practice question1 Python program3 problems1 Python lab
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Lesson 2about 25 min
Properties of nCr
Choosing $r$ to take is choosing $n - r$ to leave, so $^{n}C_{r} = {}^{n}C_{n - r}$ , and $^{n}C_{a} = {}^{n}C_{b}$ means $a = b$ or $a + b = n$ . Pascal's rule, $^{n}C_{r} + {}^{n}C_{r - 1} = {}^{n + 1}C_{r}$ , builds Pascal's triangle row by row. Combinations also count lines, diagonals and triangles in geometry.
1 simulation1 practice question1 Python program6 problems
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Lesson 3about 25 min
Selecting, then arranging
Selections often come with conditions. Fix a must-have person and choose the rest; drop a banned one and choose from the others. Choosing from separate groups multiplies the counts. For "at least one", subtract the selections with none from the total. To make words or codes, first select, then arrange with $r!$ .
1 simulation1 practice question1 Python program3 problems1 Python lab1 question set1 note
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15 questions · 20 min

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