Question

How many words, with or without meaning can be made from the letters of the word MONDAY, assuming that no letter is repeated, if all the letters are used at a time?

Answer 1

Create the number of boxes equal to the number of letters in the word MONDAY. Now, find out the total number of ways in which the first box can be filled by using the formula ${}^{n}{{C}_{r}}$, where ‘n’ is the total number of letters available and ‘r’ is the number of ways in which remaining boxes can be filled. Multiply all the results to get the answer.

Complete step-by-step solution
We have to find the number of words that can be formed by using the word MONDAY.
The total number of letters in the word MONDAY = 6.
So, let us create six boxes in which these letters are to be filled in different ways.
Now, the first box can be filled with either of the 6 letters of the word MONDAY. So, we have to select one letter from these 6 letters. Therefore, Number of ways to select 1 letter from 6 letters = ${}^{6}{{C}_{1}}=6$.
Now, 5 letters are left and 5 boxes are left. So, the second box can be filled with either of the 5 letters. Therefore, Number of ways to select 1 letter from remaining 5 letters = ${}^{5}{{C}_{1}}=5$.
Similarly, the third box can be filled with either of the remaining 4 letters. Therefore, Number of ways to select 1 letter from remaining 4 letters = ${}^{4}{{C}_{1}}=4$.
For the fourth box, 3 letters are remaining. Therefore, Number of ways to select 1 letter from remaining 3 letters = ${}^{3}{{C}_{1}}=3$.
For the fifth box, we have 2 letters remaining. Therefore, the number of ways to select 1 letter from these 2 letters = ${}^{2}{{C}_{1}}=2$.
So, the last box can be filled with the remaining 1 letter, and therefore, the number of ways to select this letter is 1.
Now, total number of ways in which all the boxes are filled,
= 6 $\times$ 5 $\times$ 4 $\times$ 3 $\times$ 2 $\times$ 1
= 6!
= 720

Note: One may note that we have to fill all the boxes, so in the end, we need to multiply all the individual ways of filling the boxes. Never add all the individual ways in such cases. It is better to draw the number of boxes equal to the number of letters in the word because it makes our visualization easy and the chances of getting confused are less.