SaitechAI Worksheet — Probability Involving Combinations

IIT JEE focus • Concepts + Worked Examples + Target Practice (6 problems) • Keys hidden until Submit

Order ignored → use nCr Equally likely outcomes Complement method

Concept Notes

In many IIT JEE probability questions, you count outcomes using combinations when the order of selection does not matter. Typical contexts: card hands, committees, choosing balls together.

1) Combination basics

Formula

nCr = n! / (r! (n-r)!)

Use combinations when selection is unordered (AB = BA). Keywords: “choose”, “select”, “hand of cards”, “committee”.

2) Probability via counting

Main rule

P(E) = (Number of favourable combinations) / (Total number of possible combinations)

Model A — Cards (few examples)

Example A1: 2 aces in 2-card draw (from 52)

Total = 52C2, Favourable = 4C2 → P = (4C2)/(52C2)

Reason: a 2-card selection is unordered, so combinations.

Example A2: At least 1 ace in 2 cards

P(at least 1 ace) = 1 − P(no ace) = 1 − (48C2)/(52C2)

Complement is faster than counting “exactly 1 ace” + “2 aces”.

Model B — Balls from a bag (few examples)

Example B1: 2 red balls

Bag: 5 red, 4 blue. Two drawn together.

Total = 9C2, Favourable = 5C2 → P = (5C2)/(9C2)

Example B2: Exactly 1 red and 1 blue

Total = 9C2, Favourable = (5C1)(4C1) → P = (5C1·4C1)/(9C2)

Model C — Committees / selection (few examples)

Example C1: Exactly 1 woman in a 3-member committee

From 6 men and 4 women.

Total = 10C3, Favourable = (4C1)(6C2) → P = (4C1·6C2)/(10C3)

Example C2: At least 1 woman in a 3-member committee

P = 1 − P(no woman) = 1 − (6C3)/(10C3)

Quick IIT JEE tips

Write denominator first (total combinations).
If “at least / at most” appears, try the complement method.
Use combinations when order is irrelevant; avoid permutations unless arrangement is asked.
Final probability must be between 0 and 1.

Target Practice (TP) — Worksheet (6 Problems)

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Name

Class

Date

Total Marks: 12

6 × 2 marks

Keys hidden until submit

Q1.

2 marks

From a standard 52-card deck, 2 cards are drawn together. Find the probability that both are kings.

Key (Q1)

Total outcomes = 52C2
Favourable = 4C2 (choose 2 kings from 4)
P = (4C2)/(52C2)

Q2.

2 marks

A bag contains 7 white and 5 black balls. Two balls are drawn together. Find the probability that exactly one ball is black.

Key (Q2)

Total = 12C2
Favourable = (5C1)(7C1)
P = (5C1·7C1)/(12C2)

Q3.

2 marks

From 10 students, a team of 3 is formed. Find the probability that two particular students are both included in the team.

Key (Q3)

Total teams = 10C3
Favourable: include the 2 fixed students, choose 1 more from remaining 8 → 8C1
P = (8C1)/(10C3)

Q4.

2 marks

A committee of 4 is chosen from 6 men and 5 women. Find the probability that the committee has at least one woman.

Key (Q4)

Total = 11C4
P(at least 1 woman) = 1 − P(no woman)
No woman: choose all 4 from 6 men → 6C4
P = 1 − (6C4)/(11C4)

Q5.

2 marks

From a 52-card deck, 3 cards are drawn together. Find the probability that all 3 are spades.

Key (Q5)

Total = 52C3
Favourable: spades = 13 cards → 13C3
P = (13C3)/(52C3)

Q6.

2 marks

A box contains 4 defective and 16 good items. If 2 items are chosen at random, find the probability that at least one is defective.

Key (Q6)

Total = 20C2
P(at least 1 defective) = 1 − P(no defective)
No defective: choose both from 16 good → 16C2
P = 1 − (16C2)/(20C2)

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