Permutations and combinations for NET: counting rules and worked MCQs

Permutations and combinations on NET test whether you can tell an arrangement from a selection, and then apply nPr, nCr and a handful of standard results. It is one of the highest-yield maths chapters, and nearly every question is a known type: diagonals, lines through points, committees, digits, words with repeated letters.

Key facts

• Mathematics is 50% of NET-Engineering and NET-Natural Sciences, and 30% of NET-Architecture
• Permutations and Combinations is unit 7 of the Class 11 PECTAA mathematics book (National Curriculum 2023)
• Our chapter map estimates about 9 MCQs from this chapter, among the highest of any maths chapter. NUST does not publish chapter counts

Source: NUST subject weightings, verified 22 August 2026; SailToNUST chapter map

What the chapter covers#

The chapter builds counting from factorials up. Permutations count arrangements, where order matters. Combinations count selections, where order does not. On top of those sit the selection rules: always include, never include, at least, at most, never together.

Our practice set for this chapter also includes basic probability, since most probability questions are counting questions divided by a total.

The first question to ask on every MCQ: does swapping two chosen items give a new outcome? If yes, it is a permutation. If no, it is a combination.

Formulas#

n!   = n(n - 1)(n - 2) ... 1,        0! = 1
nPr  = n! / (n - r)!
nCr  = n! / [r! (n - r)!]
nCr  = nC(n - r)
nCr + nC(r - 1) = (n + 1)Cr

Arrangement rules:

n items with p alike, q alike, ...     n! / (p! q! ...)
n items in a circle                    (n - 1)!
necklace or garland (can flip)         (n - 1)! / 2
two items always together              treat as one block: (n - 1)! x 2!
two items never together               total - together

Geometry counts:

diagonals of an n-sided polygon        nC2 - n = n(n - 3)/2
lines from n points, none collinear    nC2
lines from n points, m collinear       nC2 - mC2 + 1
triangles from n points, m collinear   nC3 - mC3
rectangles on an 8 x 8 chessboard      9C2 x 9C2

Selections:

at least one from n distinct items     2^n - 1
probability of an event                favourable outcomes / total outcomes

Worked MCQs#

Five standard types from our practice bank. We counted each one by hand and checked the small cases.

1. How many diagonals does a 7-sided polygon have?

7(7 - 3)/2 = 14

2. 13 points lie in a plane. No three are collinear except 5 of them. How many lines can be drawn?

Count as if none were collinear, remove the lines among the 5, then add back the one line they all lie on:

13C2 - 5C2 + 1 = 78 - 10 + 1 = 69

3. How many 5-digit numbers can be made from 1, 2, 3, 4, 5 (no repeats) with 1 and 5 never next to each other?

total arrangements          5!        = 120
1 and 5 together            4! x 2!   = 48
never together              120 - 48  = 72

4. How many rectangles are there on a chessboard?

A rectangle is fixed by choosing 2 of the 9 horizontal lines and 2 of the 9 vertical lines:

9C2 x 9C2 = 36 x 36 = 1296

5. A team of 11 is chosen from 13 players: 4 bowlers, 2 wicketkeepers and 7 others. The team must have exactly one wicketkeeper and at least 3 bowlers. How many ways?

One keeper takes one place, leaving 10 places for bowlers and others. Split by the number of bowlers:

3 bowlers + 7 others:  4C3 x 7C7 = 4 x 1 = 4
4 bowlers + 6 others:  4C4 x 7C6 = 1 x 7 = 7
choose the keeper:     2C1 = 2
total = (4 + 7) x 2 = 22

Common traps#

• Order or not. "Arrange", "form numbers", "signals" and "words" mean order matters. "Choose", "select", "committee" and "team" mean it does not.
• The order inside a block. In "never together" questions, students subtract the together case but forget the 2! for the internal order of the pair.
• Repeated letters. ADVANCE has two A's. Arrangements of all 7 letters are 7!/2! = 2520, not 5040.
• "Any number of flags". With 3 different flags, signals using 1, 2 or 3 flags give 3 + 6 + 6 = 15. Students count only the full arrangements.
• The +1 in collinear lines. In example 2, subtracting 5C2 removes the collinear line completely. You must add it back once.

How to prepare#

Do this chapter by question type, not by formula. Spend one session each on geometry counts, arrangements with restrictions, and committee selections. Then do a mixed timed set, because the hard part on the paper is recognising the type.

The AP sum from sequences and series is the same as nC2, and the binomial theorem chapter uses nCr for every coefficient. The NET mathematics guide explains why chapters with high typical counts, like this one, deserve extra time.

FAQ#