Question

Find the number of words that can be formed using the letters of the word WATCH. If these words are organized in a dictionary, what is the ${{50}^{th}}$ word?

Answer 1

Use the fact that the number of ways in which n distinct objects can be arranged in n different places is given by $n!$. Hence find the total number of words that can be formed using the letters of the word WATCH. Hence find the word whose rank is 50.

Complete step-by-step solution:
The letters in the word WATCH are A, C, H, T, W. All are distinct.
We know that the number of ways in which n distinct objects can be arranged in n different places is given by $n!$
Hence the number of arrangements of the letters of the word WATCH is equal to $5!=120$
Hence the total number of words that can be formed using the letters of the word WATCH is 120.
Now the number of words starting with A is equal to the number of arrangements of the letters C, H, T, W which is equal to $4!=24$
The number of words starting with C is equal to the number of arrangements of the letters A, H, T, and W which is equal to $4!=24$
Hence HACTW is ${{\left( 24+24+1 \right)}^{th}}={{49}^{th}}$ word in the dictionary
Hence, HACWT is the ${{50}^{th}}$ word in the dictionary.
Hence the rank of word HACWT is 50.

Note: In these types of questions we need to arrange the letters in increasing alphabetical order and try finding the position of each letter in the given rank. As done above, we found that there is 24 letter each starting with A and C. Hence all the letters starting with A and C will occupy the first 48 positions. This leaves H occupying the first place and so on.