Respuesta :
Answer:
- If repetition of letters is allowed = 343 words
- If repetition of letters is not allowed = 120 word
Step-by-step explanation:
We can find the number of 3-letter words that can be made from the letters WASTEFUL using possibilities and permutation.
(1) If repetition of letters is allowed, then each of the 3 letters will have 7 outcomes:
- 1st letter has 7 outcomes → W, A, S, T, E, F, U and L
- 2nd letter has 7 outcomes → W, A, S, T, E, F, U and L
- 3rd letter has 7 outcomes → W, A, S, T, E, F, U and L
Therefore, the number of outcomes = [tex]7\times7\times7[/tex]
= [tex]\bf343\ words[/tex]
(2) If repetition of letters is not allowed, then we can find the outcome using this permutation formula:
[tex]\boxed{_nP_r=\frac{n!}{(n-r)!} }[/tex]
where:
- n = total number of objects
- r = total number of selected objects
Given:
- n = 7
- r = 3
[tex]\displaystyle _nP_r=\frac{n!}{(n-r)!}[/tex]
[tex]\displaystyle =\frac{7!}{(7-3)!}[/tex]
[tex]\displaystyle =\frac{7!}{4!}[/tex]
[tex]=7\times6\times5[/tex]
[tex]=\bf210\ words[/tex]
With repetition allowed, there can be 512 possible 3-letter words made from the letters of WASTEFUL. Without repetition, there are 336 possible words. This calculation is based on the principles of permutations.
The question asks how many 3-letter words can be created from the letters of the word WASTEFUL with and without repetition of letters. To find the answer, we use the principles of permutations and combinations.
With Repetition Allowed
When repetition is allowed, we can fill each of the three positions by any of the 8 different letters (W, A, S, T, E, F, U, L). Therefore, for each position, there are 8 possibilities:
First letter: 8 options
Second letter: 8 options
Third letter: 8 options
By multiplying these together (8 x 8 x 8), we get a total of 512 possible 3-letter words.
Without Repetition Allowed
When repetition is not allowed, the number of options decreases with each letter because each letter can only be used once:
First letter: 8 options
Second letter: 7 options (since one letter has been used)
Third letter: 6 options (because two letters are already used)
Again, we find the total by multiplying the number of options for each position (8 x 7 x 6), which gives us 336 possible 3-letter words without repetition.
