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Permutations and Combinations - Previous Year Questions

⏱️ Est. Revision Time: 15 Mins
🔍
📅 Year 2024
Q1
CBSE Board Exam 20243 Marks
How many words (with or without meaning) can be formed using all letters of the word INDEPENDENCE such that all vowels are together?
▶ Show Detailed Solution
INDEPENDENCE: I, N, D, E, P, E, N, D, E, N, C, E → 12 letters. Vowels: I, E, E, E, E (4 E's + I = 5 vowels).
Treat the 5 vowels as one unit: 12 − 5 + 1 = 8 units, with D (×2), N (×3) repeating.
Arrangements = 8!/(2!·3!) × (vowel arrangement) 5!/4! = 3360 × 5 = 16800.
Q2
CBSE Board Exam 20243 Marks
From a committee of 8 persons, in how many ways can a chairman and a vice-chairman be chosen, assuming one person cannot hold more than one position?
▶ Show Detailed Solution
Choose chairman: 8 ways. Choose vice-chairman from remaining 7: 7 ways.
Total = 8 × 7 = 56 ways.
(Order matters: ⁸P₂ = 8!/6! = 56.)
📅 Year 2023
Q1
CBSE Board Exam 20233 Marks
A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team has (i) no girl (ii) at least 3 girls?
▶ Show Detailed Solution
(i) No girl: choose 5 boys from 7 = ⁷C₅ = 21.
(ii) At least 3 girls: 3 girls + 2 boys = ⁴C₃·⁷C₂ = 4 × 21 = 84; 4 girls + 1 boy = ⁴C₄·⁷C₁ = 1 × 7 = 7.
Total = 84 + 7 = 91 ways.