Permutations & Combinations
Arrange, select, and count — without re-counting.
1 Explore — try these first
Try before you watch. Pick a level below and give the problem an honest try on paper first — wrong turns and all. Then open the video to see the trick. Every level rides one habit: Decide order-or-not first, then reach for one named tool: factorial, permutation, or combination.
★ · Factorial · Permutation · Combination
Five students stand in a single line for a welcome photo. How many different lineups are possible?
L0 · Factorial Lineup
Five students stand in a single line for the welcome photo. How many different lineups are possible?
L1 · Block Trick
Six students stand in a welcome line. Anna and Ben (buddies) must stand next to each other. Cara is holding the welcome sign, so she must stand at the front of the line. How many different lineups are possible?
L2 · Subtract the Bad
Six students in an inclusive welcome line. Anna and Ben (buddies) must stand together. Cara holds the welcome sign at the front. Dan and Erin help different groups, so they must not stand next to each other. How many lineups are possible?
L3 · Ordered Pairing
Four student helpers will each be paired one-to-one with a different new student chosen from six new students. However, Helper Anna cannot be paired with new student Xavier, because Xavier needs language support from a different helper. How many different buddy pairings are possible?
L4 · Group Combinations
An inclusive student support committee of 4 students will be chosen. There are 3 mentors, 3 club leads, and 3 athletes (9 students total). Each of the three groups must be represented on the committee. However, Anna (a mentor) and Zane (an athlete) cannot both be chosen. How many different committees are possible?
L5 · Quotas + Exclusion
An inclusive outreach team of 6 students will be chosen from 3 mentors, 3 club leads, and 3 athletes. The team must include at least 2 mentors, at least 1 club lead, and at least 2 athletes. Anna (a mentor) and Yara (a club lead) cannot both be chosen. How many different teams are possible?
2 Learn — watch the solutions
Gave each one a real try? Now watch the trick. (Stuck is fine — that's the point.)
★ · Permutations vs Combinations
Peek the trick — Swap-Test Decision
Ask: if I swap two of the chosen items, is it the same answer? If yes, order doesn't matter — use a combination; if no, order matters — use a permutation (or a factorial when you arrange all of them).
L0 · Factorial Lineup
Peek the trick — Multiply Down to 1
To line up n distinct people in a row, fill the first spot in n ways, the next in n−1, and so on down to 1. The total is n! arrangements.
L1 · Block Trick
Peek the trick — Treat the pair as one block, then multiply by 2
When two specific people MUST stand together, glue them into one block, arrange the (n−1) units in a line, then multiply by 2 for the internal swap of the pair.
L2 · Subtract the Bad
Peek the trick — Count all, then subtract the together-cases
When two specific people must NOT stand together, count ALL arrangements first, count the BAD ones where they ARE together using the Block Trick, and subtract.
L3 · Ordered Pairing
Peek the trick — P(n, r) — and subtract P(n−1, r−1) for a forbidden pairing
When you assign r distinct slots to people picked from n, use P(n, r) = n × (n−1) × … × (n−r+1). To exclude one specific forbidden pairing, lock that pair in place and subtract P(n−1, r−1).
L4 · Group Combinations
Peek the trick — Enumerate valid group splits, multiply C(n_i, k_i), then sum
When a committee of size k is drawn from several groups and each group must be represented, list every valid split of how many come from each group, multiply the C(n_i, k_i) for each split, and add the products.
L5 · Quotas + Exclusion
Peek the trick — List splits that satisfy every minimum, sum the C-products
When each group has a minimum and the total team size has extras left to distribute, list every valid split (minimums plus extras), multiply the C(n_i, k_i) for each split, and add them all together.
3 Master — practice on your own
Print the practice sheet and solve without the videos. Check your answers at the back — if one is wrong, the answer key names the trick so you know exactly which video to rewatch.
Download fresh practice problems PDF
