Stars & Bars (Care Packs)
Share identical things across distinct rooms — one toolkit, six tricks.
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: Reserve every required minimum first, then count what is left as stars between bars.
★ · Factorial · Permutation · Combination
Four friends — Ada, Ben, Cy, and Dee — want to send TWO of them to represent the class. How many different pairs of representatives are possible?
L0 · Reserved Seats
A school is sharing 4 identical care packs among 3 support rooms — the Quiet Room, the Newcomer Room, and the Reading Corner. Each room must get at least 1 care pack. How many different distributions are possible?
L1 · Stars and Bars
A school is sharing 7 identical care packs among 3 support rooms. Each room must get at least 1 care pack. How many different distributions are possible?
L2 · Pick One, Count the Rest
A school is sharing 10 identical care packs among 3 support rooms — Quiet, Newcomer, and Reading. Each room must get at least 1 care pack. The Quiet Room must receive more care packs than the Reading Corner. How many different distributions are possible?
L3 · Reserve, Then Bars
A school is sharing 10 identical care packs among 4 support rooms — Quiet, Newcomer, Reading, and Wellness. The Quiet Room must receive at least 3 care packs. Each other room must receive at least 1 care pack. How many different distributions are possible?
L4 · Symmetry Halves
A school is sharing 14 identical care packs among 4 support rooms. Each room must receive at least 2 care packs. The Quiet Room must receive more care packs than the Newcomer Room. How many different distributions are possible?
L5 · Choose, Then Pour
A school is sharing 14 identical care packs among 4 support rooms. Each room must get at least 2 and at most 6 care packs. Exactly two of the four rooms must end with an odd number of care packs. How many different distributions are possible?
2 Learn — watch the solutions
Gave each one a real try? Now watch the trick. (Stuck is fine — that's the point.)
★ · What Is Choosing
Peek the trick — Choose Shortcut
When order does not matter and you are picking k items from n, compute n choose k as the top k of n multiplied together, divided by k down to 1. No factorials, no calculator.
L0 · Reserved Seats
Peek the trick — Reserved Seats
When every room must get at least one of the identical items, hand out one to each room first. Then only count the ways to share what is left over.
L1 · Stars and Bars
Peek the trick — Stars and Bars
Lay the leftover items in a row as stars, then place R minus 1 bars between them as dividers. Every arrangement is a different distribution, so count by choosing the bar positions: n plus R minus 1 choose R minus 1.
L2 · Pick One, Count the Rest
Peek the trick — Pick One, Count the Rest
When one room has an ordering or inequality constraint, fix that room's count first. For each value, count the unconstrained rest with stars and bars. Add the case counts.
L3 · Reserve Then Bars
Peek the trick — Reserve, Then Bars
Reserve every room's mandatory minimum first — even when minimums differ by room. Then apply stars and bars to the leftover items: extras plus R minus 1 choose R minus 1.
L4 · Symmetry Halves
Peek the trick — Symmetry Halves
When a problem has a strict inequality between two rooms, count ALL distributions first. Subtract the cases where the two are equal. By symmetry the rest splits evenly into the two strict directions, so divide by 2.
L5 · Choose Then Pour
Peek the trick — Choose, Then Pour
When the constraint picks out a SUBSET of the rooms, choose which rooms first using R choose k. Then for each choice count the valid distributions and multiply.
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
