Given the word OCTOBER, how many different 3 letter words can be created??
we may group the word as:(O O),c,t,b,e,r and consider the cases where the three letter words are all distinct and alike of a particular kind since the sizes of the words we want is less than the size of the given word. that is 3<7.case 1:if the letters are all distinct,then three letters can be selected from six different letters in 6c3 ways.the three letters can then be ordered in 3!ways after being selected and by the product rule this gives 6c3.3!=120 ways.case 2:if the three letters we want may be alike of a particular kind.that is forming words like (oto),(oor),(oob) and so on.then the two o's can be selected in one way while the remaining one letter can be selected from either c or t or b or e or r in 5c1 ways also the three letter words can be ordered in 3!ways after being selected the result will be divided by 2! since we have two o's and by the product rule this gives 1.5c1.3!/2!=15 ways.so adding the two cases gives the number of words we want.case 1+case 2=120+15=135ways.
there are 7 letters, so there are
7P3 = 210 ways to choose 3 of them.
But the two O's are indistinguishable, so there are only
7P3/2! = 105 unique words
To find the number of different three-letter words that can be created from the word "OCTOBER," we can use the concept of combinations.
First, let's identify the number of options we have for each position in the three-letter word.
In this case, we have seven unique letters: O, C, T, B, E, R. However, the letter "O" appears twice in the word "OCTOBER." Therefore, we need to consider the number of distinct letters for each position:
- For the first letter, we have 7 options (O, C, T, B, E, R).
- For the second letter, we have 6 options (excluding the letter chosen for the first position).
- For the third letter, again we have 6 options (since we still have 7 unique letters, but we exclude the letter chosen for the first position as well as the letter chosen for the second position).
To find the total number of different three-letter words, we multiply the number of options for each position together:
7 * 6 * 6 = 252
Therefore, there are 252 different three-letter words that can be created from the word "OCTOBER."