Question

How many four lettered words can be made with the word FLOWER?

Answer 1

In the above question we have a word FLOWER which has distinct letters. We will use the formula of combination to find the number of 4 lettered words which can be formed with the word FLOWER.

Complete step-by-step answer:
FLOWER has all the distinct letters i.e. (F, L, O, W, E, R); So, words which will form can’t have repeated letters.
Now, FLOWER has 6 letters out of which we have to take four letter words. To choose four lettered words we will use $^6{C_4}$ and to arrange these four letters we use 4!
Four lettered words can be made with the word FLOWER ${ = ^6}{C_4} \times 4!$
$= \dfrac{{6!}}{{4! \times 2!}} \times 4!$
[Combination Formula: Number of ways to choose r letters from n letters is given by ]
$= \dfrac{{6 \times 5 \times 4 \times 3 \times 2 \times 1}}{{2 \times 1}} = 360$ words
Hence, the number of four lettered words can be made with the word FLOWER is 360 words.

Note: If we have repeated letters in the given word then we would have different conditions to arrange them. But in this question, we have distinct words so we do not need to use these conditions. Permutation and Combination is used to represent a group of objects by selecting them from a set. Permutation is used to arrange the objects into order or a sequence. The number of permutations of n objects taken ‘r’ at a time is determined by the following formula: $P\left( {n,r} \right) = \dfrac{{n!}}{{\left( {n - r} \right)!}}$. Combination is used to select items from a collection, where order of selection does not matter. The number of combinations of n objects taken ‘r’ at a time is determined by the following formula $C(n,r) = \dfrac{{n!}}{{\left( {n - r} \right)!r!}}$.