Counting
Permutation and combination questions turn on a single question: does the order matter? If it does, it is a permutation, nPr. If it does not, it is a combination, nCr, which is the same count divided by r factorial. Arranging people in seats is a permutation; picking a committee is a combination.
Counting questions carry one or two marks in prelims and appear more often in mains, and they are the topic where the wrong formula produces a wrong answer with total confidence. Because nPr is always r! times nCr, choosing wrongly is never close — and both values are always in the options.
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If Ram-then-Sita differs from Sita-then-Ram, order matters and it is a permutation. If picking Ram and Sita is the same set either way, it is a combination. This one test resolves the majority of questions before any arithmetic.
Arrange, seat, rank, form a number and password all mean order matters. Select, choose, committee, team and handshake mean it does not. The vocabulary is consistent enough across papers to be relied on.
Arranging the letters of a word with repeated letters divides the factorial by the factorial of each repeat count. For a five-letter word with one letter twice, that is 5! ÷ 2!. Forgetting the division doubles the answer.
If three people must sit together, bundle them into a single block, arrange the blocks, then multiply by the arrangements inside the block. For n items with a block of three, that is (n − 2)! × 3!.
| The situation | Use |
|---|---|
| Arranging r of n in order | nPr = n! ÷ (n − r)! |
| Selecting r of n, order irrelevant | nCr = n! ÷ (r!(n − r)!) |
| Arranging a word with repeats | n! ÷ (product of each repeat's factorial) |
| Round-table arrangement of n | (n − 1)! |
| A group that must stay together | Bundle as one, then multiply by its internal arrangements |
| Handshakes among n people | nC2 |
Generated by the same engine that mints the ComputePrep daily. Each answer is computed from the numbers printed in the question, and the walkthrough below each one is the engine's own working — not a solution written afterwards. Reload this page's live drill and you get different numbers.
Consider all arrangements of the letters of the word "BUTTER".
How many distinct arrangements are there?
Consider all arrangements of the letters of the word "APPLE".
How many distinct arrangements are there?
9 athletes run a race.
Gold, silver and bronze medals are awarded.
In how many ways can the medals be won?
The wrong options are not random numbers. Each one is the result of a specific careless error for this topic, so picking one tells you which habit is costing you marks.
The two differ by a factor of r!, so the error is never a near miss — and because both values appear in the options, a confident wrong answer is the usual outcome. Run the swap test before computing.
Arranging the letters of a word with a repeated letter and not dividing by 2! gives exactly twice the correct count. Scan for repeats before writing the factorial.
Seating n people around a round table is (n − 1)!, not n!, because rotations of the same arrangement are identical. Round-table questions are common precisely because this is easy to miss.
A permutation counts arrangements, where order matters; a combination counts selections, where it does not. They are related by nPr = nCr × r!. The practical test is to swap two of your chosen items and ask whether you now have a different outcome — if yes, it is a permutation.
Take the factorial of the total number of letters and divide by the factorial of each letter's repeat count. A six-letter word with one letter appearing twice and another three times gives 6! ÷ (2! × 3!). Skipping the division always overcounts.
Because a circle has no fixed starting seat, so rotating everyone by one place produces the same arrangement. Fixing one person's position removes that duplication and leaves the remaining n − 1 to be arranged freely.
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