Permutation and Combination Basics

Permutation and combination sounds intimidating but the whole topic turns on one question: does the order matter? If order matters it is a permutation; if it does not, it is a combination. SBI Clerk Prelims touches this only lightly, so you need the basic formulas and the order test — nothing deeper. Get the factorial idea and the nPr-versus-nCr distinction clear, and the occasional question that appears becomes a quick, confident mark rather than a time sink.

Factorial: The Building Block

A factorial, written n!, is the product of all whole numbers from n down to 1, so 5! = 5 × 4 × 3 × 2 × 1 = 120. By definition 0! equals 1, a fact examiners love to test. Factorials grow explosively, so you rarely compute a large one fully — instead you cancel common terms between numerator and denominator. Recognising that 7! = 7 × 6 × 5!, for instance, lets you cancel the 5! against another factorial and avoid multiplying huge numbers by hand.

Permutation vs Combination — The Order Test

A permutation counts arrangements where sequence matters, such as forming a three-digit number or seating people in a row. A combination counts selections where sequence is irrelevant, such as choosing a committee or picking fruits from a basket. The one-line test is decisive: if swapping two chosen items creates a new outcome, order matters and you use nPr; if swapping changes nothing, order does not matter and you use nCr. Words like arrange, order and rank signal permutation; select, choose and committee signal combination.

The Two Formulas

For permutations, nPr equals n factorial divided by (n minus r) factorial, counting the ways to arrange r items from n. For combinations, nCr equals n factorial divided by r factorial times (n minus r) factorial, counting the ways to select r items from n. The link between them is worth memorising: nPr equals nCr times r factorial, because every selection of r items can itself be arranged in r factorial orders. Two handy facts — nC0 = nCn = 1 and nCr = nC(n−r) — often let you replace a big r with a small one.

nPr = n!/(n−r)! nCr = n!/[r!(n−r)!] nCr = nC(n−r) nP r = nCr × r!

Solved Examples

Example 1: Ways to arrange 3 of 5 books on a shelf = 5P3 = 5 × 4 × 3 = 60, since order on a shelf matters. Example 2: Ways to choose 3 of 5 books to gift = 5C3 = 5C2 = (5 × 4)/(2 × 1) = 10, using nCr = nC(n−r) to shrink the arithmetic.

✗ Using nPr to select a committee where order is irrelevant  |  ✓ Applying the order test first, then choosing nCr

Speed Shortcuts for nCr

You almost never need the full factorial formula for nCr in a Prelims question. Instead, write nCr as a short falling product: nC2 is n times (n minus 1) over 2, and nC3 is n times (n minus 1) times (n minus 2) over 6. Because the exam keeps r small, this compact form answers most questions in one line without touching a factorial. Pair it with the symmetry rule nCr = nC(n−r) to convert an awkward large r into a tiny one — 10C8 becomes 10C2, which is just (10 × 9)/2 = 45. Between the falling-product form and the symmetry swap, you can evaluate any combination the exam is likely to ask entirely in your head, which is exactly the kind of speed the sectional timing rewards.