Permutations & Combinations
Permutation and Combination
star_batch_jee_advanced_2025
Grade 11
Question:
A contest consists of ranking 10 songs of which 6 are Indian classic and 4 are western songs. Number of ways of ranking so that (mention correct statements)
There are exactly 3 Indian classic songs in top 5 is $(5!)^3$
Top rank goes to Indian classic song is $6(9!)$
The rank of all western songs are consecutive is $4!7!$
The 6 Indian classic songs are in a specified order is $^{10}P_6$
Step-by-Step Solution
Key Concept: Each constraint reduces the problem to placing restricted groups of songs while maintaining their relative arrangements or fixed positions.
We verify each statement: (1) Exactly 3 Indian songs in top 5: Choose 3 from 6 Indian songs in $\binom{6}{3}$ ways, choose 2 from 4 Western songs in $\binom{4}{2}$ ways, arrange these 5 in top 5 in $5!$ ways, arrange remaining 5 in bottom 5 in $5!$ ways = $\binom{6}{3}\binom{4}{2}(5!)(5!) = 20 \times 6 \times (5!)^2 = (5!)^3$ ✓ (2) Top rank Indian: Fix 1 Indian song at rank 1 in 6 ways, arrange remaining 9 songs in $9!$ ways = $6(9!)$ ✓ (3) All Western consecutive: Treat 4 Western songs as one block, arrange this block + 6 Indian songs = 7 objects in $7!$ ways, arrange 4 Western songs within block in $4!$ ways = $4! \times 7!$ ✓ (4) 6 Indian songs in specified order: Treat these 6 as one unit in fixed order, choose 6 positions from 10 for them in $\binom{10}{6}$ ways = $^{10}P_6$ ✓
Correct Answer: 1,2,3,4