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CBSE · Class 9 · Mathematics

Combinatorics (Advanced)

Standard difficulty · 20 questions · 40 marks · Time: 32 minutes

Name: Date: Score: ____ / 40

Show your working for Sections B and C. Answers are on the last page — try every question first!

Section A — Choose the correct answer8 × 1 = 8 marks
1.
In a race of 6 runners, in how many ways can Gold, Silver and Bronze be awarded (no ties)?
  1. 216
  2. 20
  3. 30
  4. 120
2.
In how many ways can a panel of 5 be chosen from 15 people?
  1. 75
  2. 1,365
  3. 3,003
  4. 360,360
3.
0! is taken to be:
  1. 10
  2. −1
  3. 0
  4. 1
4.
If n! ÷ (n − 2)! = 182, then n is:
  1. 14
  2. 15
  3. 13
  4. 16
5.
How many 4-digit numbers greater than 4,000 can be formed from the digits 3, 4, 6, 7, 8, 9 with no digit repeated?
  1. 1,080
  2. 360
  3. 300
  4. 240
6.
Seating people in a row of chairs is a problem of:
  1. sets
  2. permutations
  3. combinations
  4. factors
7.
11 points lie in a plane, and exactly 3 of them are on one straight line (no other three are). How many different straight lines pass through at least two of them?
  1. 53
  2. 29
  3. 55
  4. 52
8.
A dhaba offers 5 starters, 2 main dishes and 5 desserts. A guest orders exactly one dish in all. In how many ways?
  1. 12
  2. 50
  3. 15
  4. 11
Section B — Short answer8 × 2 = 16 marks
9.
Find x if 1/8! − 1/9! = x/10!.
10.
How many 5-letter words (with or without meaning) can be formed from the letters of PLANET if no letter is repeated?
11.
A paper has 8 questions in Part A and 4 in Part B. A student must answer 6 questions, at least 2 from each part. In how many ways can the questions be chosen?
12.
At a party everyone shook hands with everyone else exactly once. There were 15 handshakes. How many people were there?
13.
6 different Mathematics books and 4 different Science books are to be arranged on a shelf so that the Science books are always together. In how many ways?
14.
How many 3-digit even numbers can be formed from the digits 1, 3, 6, 7, 9 if no digit is repeated?
15.
If ⁿC3 = ⁿC2, find n and ⁿC3.
16.
1 × 1! + 2 × 2! + … + 10 × 10! = x! − 1. Find x.
Section C — Word problems4 × 4 = 16 marks
17.
A committee of 5 is to be chosen from 8 men and 5 women so that it has at least 4 women. In how many ways can this be done?
18.
In how many ways can the letters of MOTHER be arranged so that (a) all the vowels come together (b) the vowels do not all come together?
19.
A door code has 5 digits, each one of the 10 digits 0 to 9. How many codes are possible if (a) digits may repeat (b) no digit repeats (c) no digit repeats and the first digit is not 0?
20.
There are 14 points in a plane; exactly 9 of them lie on one straight line and no other three are in a line. Find (a) the number of straight lines through two or more of the points (b) the number of triangles with corners at the points.
AmbifulWorksheets
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CBSE · Class 9 · Mathematics

Answer key

Combinatorics (Advanced) · AW-9M-ACB-2026-V1 · For parents — fold back or tear off before handing the sheet to your child.

QAnswerWorking
1(d) 1206P3 = 6 × 5 × 4
2(c) 3,00315C5 = 360,360 ÷ 5!
3(d) 1so that formulas like ⁿCn = n! ÷ (n! × 0!) = 1 work
4(a) 14n! ÷ (n − 2)! = n(n − 1) = 14 × 13
5(c) 300the thousands digit must be 4 or larger (5 choices); the other 3 places: 5P3 = 60; 5 × 60
6(b) permutationsthe order matters
7(a) 5311C2 − 3C2 + 1 = 55 − 3 + 1
8(a) 12OR: 5 + 2 + 5
9x = 80multiply by 10!: x = 10 × 9 − 10 = 80
10720PLANET has 6 different letters: 6P5 = 6 × 5 × 4 × 3 × 2
116728C2 × 4C4 + 8C3 × 4C3 + 8C4 × 4C2
126ⁿC2 = n(n − 1)/2 = 15, so n = 6
13120,960treat the Science books as one block: 7! × 4! = 5,040 × 24
1412by units digit: 6 → 12
15n = 5; 10ⁿCᵣ = ⁿCn−ᵣ, so n = 3 + 2 = 5; 5C3 = 10
16x = 11each term r × r! = (r + 1)! − r!; the sum telescopes to 11! − 1!
17415C4 × 8C1 + 5C5 × 8C0
18(a) 240; (b) 480(a) 4 consonants + 1 vowel block: 5! × 2!; (b) 6! − (a) = 720 − 240
19(a) 100,000; (b) 30,240; (c) 27,216(a) 105; (b) 10P5; (c) 9 × 9P4
20(a) 56; (b) 280(a) 14C2 − 9C2 + 1; (b) 14C3 − 9C3