What it does
Counting problems are hard because it is easy to count something twice or miss a case. This page lets you see every case while the numbers are small, so the formulas stop being magic:
- Arrange (order matters). Seat r of n friends on r chairs in a row. The boxes show how many choices each chair has: n, then n − 1, and so on. Multiplying them gives nPr = n!/(n − r)!, and the page lists every seating in dictionary order.
- Choose (order does not matter). Pick a team of r from n players. AB and BA are now the same team, and every team turns up r! times in the Arrange list, so nCr = nPr / r!. Every team is listed.
- Letters of a word. Type any word. If letters repeat, swapping two identical letters makes no new word, so the count is n!/(p! q! …), where p, q, … are how many times each repeated letter appears. BANANA gives 60, MISSISSIPPI gives 34,650.
The page always shows nPr, nCr and r! together, with the link nPr = nCr × r! filled in. It follows the Class 11 chapter permutations and combinations and is handy for Class 12 probability, where counting outcomes comes first.
The same nCr numbers build Pascal’s triangle in binomial expansion, and counting outcomes is the first step of a probability in sample space builder and probability tree.
How to use it
- Pick Arrange, Choose or Letters of a word at the top.
- For Arrange and Choose, set n (how many different things) and r (how many are used) with the sliders. The number boxes take n up to 30; the list then shows the first 240 cases and the formula does the rest.
- For words, type a word or tap one of the examples. Only letters count; spaces are ignored.
- Read the boxes picture, the three facts and the formulas with your numbers, then scroll through the list.
A worked example
The page starts on Arrange with n = 5 and r = 3: five friends A to E and three chairs.
- Chair 1 has 5 choices, chair 2 has 4, chair 3 has 3: 5P3 = 5 × 4 × 3 = 60. The list starts ABC, ABD, ABE, ACB, …
- Switch to Choose: the team {A, B, C} was counted 3! = 6 times above (ABC, ACB, BAC, BCA, CAB, CBA), so 5C3 = 60 / 6 = 10.
- Switch to Letters of a word with BANANA: 6 letters, A three times and N twice. 6! = 720, and 720 / (3! × 2!) = 720 / 12 = 60 different words, all listed.
Tips
- Ask “does swapping two of them make something new?” If yes, use permutations; if no, use combinations.
- nCr = nC(n − r): choosing 3 players to play is the same as choosing the 2 who sit out.
- 0! = 1, so nP0 = nC0 = 1: there is exactly one way to do nothing.
Check yourself
Four fresh questions each time: a race finish (permutation), choosing books (combination), arranging the letters of a word with repeats, and a committee made of two groups. Type a whole number and press Check. Nothing is saved.
FAQ
Why stop listing at 240? Lists grow very fast: 10P5 is already 30,240. Seeing the first few is enough to see the pattern; the formula counts the rest exactly.
Can I count bigger cases? Yes, up to n = 30 here. For big numbers, Stirling numbers, partitions and more, use the combinatorics calculator.
Is this from the NCERT book? It covers the same topics with fibub’s own examples. fibub is not part of NCERT.
Where the numbers come from
- Standard counting results: the fundamental principle of counting, n!, nPr = n!/(n − r)!, nCr = n!/(r!(n − r)!) and arrangements of n objects with repeats, n!/(p! q! …). Written for fibub.
- Checked against Python's math.perm, math.comb and itertools (every n up to 8 listed and counted) and the known count 34,650 for MISSISSIPPI.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.