How many 4 - letter words (with or without meaning) containing two vowels can be constructed using only the letters (without repetition) of the word 'LUCKNOW'?

Concept:

If n is a positive integer and r is a whole number, such that r < n, then P(n, r) represents the number of all possible arrangements or permutations of n distinct objects taken r at a time.

It can be represented as ⁿP_r = \(\frac{n!}{(n-r)!}\).

The combination is defined as “An arrangement of objects where the order in which the objects are selected does not matter.” 

ⁿC_r = \(\frac{n!}{r!(n-r)!}\) ,  when n < r

Where n = distinct object to choose from

C = Combination

r = spaces to fill

Calculation:

Vowels = 2

Consonants = 5

Total Alphabets = 7

Since 4 letter words must include 2 vowels, we don't need to select them, and the rest of the 2 letters will be taken from 5 consonants.

Number of ways of selecting 2 letters from 5 consonants = ⁵C₂ = 10

Arrangement of all 4 letters will be given by 4! = 24 ways

Total number of arrangements = 5C2 × 4! = 10 × 24 = 240 ways

∴ The total number of words that can be formed is 240.