6.Avni has 8 favorite paintings but only 2 wall hooks where she can hang them.
In how many different ways can she hang the paintings?
32
56
16
64
Problem Breakdown
Avni selects 2 paintings from 8 distinct favorites and arranges them on 2 distinct hooks. First, choose 2 paintings using combinations: C(8,2) = (8 × 7)/(2 × 1) = 28 ways. Then arrange these 2 on the hooks: 2! = 2 ways. Total: 28 × 2 = 56 or directly P(8,2) = 8 × 7 = 56.
Option Analysis
- 32: Likely C(8,2) × (2!/2) = 28, ignores hook order (arrangement matters).
- 56: Correct, C(8,2) × 2! = 28 × 2 or 8 × 7, accounts for selection and permutation. [CORRECT ANSWER]
- 16: Possibly C(8,2)/1.75 or 2^4, no combinatorial basis here.
- 64: 8 × 8 = 64 or 2^6, assumes replacement or ignores distinct selection.
Quick Comparison Table
| Option | Calculation | Why Incorrect/Correct |
|---|---|---|
| 32 | C(8,2) | Selection only, misses arrangements |
| 56 | P(8,2) | Full solution: choose + arrange |
| 16 | Unclear | Too low, doesn’t fit formulas |
| 64 | 8² | Wrong: assumes repetition |
CSIR NET Relevance
This tests permutations P(n,r) = n!/(n-r)! for ordered selections, common in aptitude sections. Practice distinguishes from combinations C(n,r) = n!/[r!(n-r)!] where order irrelevant.
