Lesson 10 of 11 · 7 min
Selecting then arranging, and dictionary order
NCERT §6.4
For the closing ceremony, the art club makes 4-letter code words from 4 vowels and 5 consonants, using 2 different vowels and 2 different consonants in each. Separately, all the arrangements of RELAY are listed like a dictionary.
The lesson in notes
In short
Many problems need both steps: first choose (combinations), then arrange the chosen objects (permutations).
Words with 2 vowels and 2 consonants from 4 vowels and 5 consonants: choose in 4C2 × 5C2 = 6 × 10 = 60 ways, then arrange the 4 letters in 4! = 24 ways, giving 60 × 24 = 1440 words.
Dictionary order lists words by their first letter, then second, and so on. The letters of RELAY in alphabetical order are A, E, L, R, Y; the first word is AELRY and the last is YRLEA.
Rank of RELAY: words starting with A, E or L: 3 × 4! = 72. Then words starting R A: 3! = 6. Then words starting R E A: 2! = 2, namely REALY and REAYL. RELAY comes next, so its rank is 72 + 6 + 2 + 1 = 81.
The same block-by-block counting answers 'how many numbers are greater than …' questions: count by the first digit, then the next, fixing one place at a time.