Simulation · Maths · Class 11
Glue them together, or keep them apart
From the lesson Arrangements with conditions in Permutations and Combinations. Change the values and watch what happens.
Glue them together, or keep them apartMaths · Class 11
The idea behind it
NCERT §6.3.3
- A leading zero makes a number shorter, so it must be removed. Three-digit numbers from the digits 0, 1, 2, 3, 4 without repetition: all arrangements 5P3 = 60, minus those starting with 0, 4P2 = 12, leaves 48. Directly: 4 × 4 × 3 = 48.
- Fixed places first: four-digit numbers greater than 3000 from the digits 1 to 5 without repetition must start with 3, 4 or 5, giving 3 × 4P3 = 3 × 24 = 72.
- Even three-digit numbers from 1, 2, 3, 4, 5 without repetition: the units digit is 2 or 4 (2 ways), then 4 and 3 ways for the other places, so 2 × 4 × 3 = 24.
- Together: tie the objects that must stay together into one block, arrange the block with the others, then arrange inside the block. RELAY with its vowels E and A together: 4 units in 4! = 24 ways, times 2! inside the block, gives 48.
- Not together: subtract from the total. RELAY has 5! = 120 arrangements, so the vowels are apart in 120 − 48 = 72 of them.
- Apart (gap method): seat 4 girls first in 4! = 24 ways; they leave 5 gaps, including the two ends. Seat the 3 boys in 3 of these gaps in 5P3 = 60 ways. No two boys are then together, and the total is 24 × 60 = 1440.