For PLANETS, find the number of :
4-letter words which must include a L and A
For MAMMALS, find the munber of:
a) 7-letter words
b) 7-letter word beggining with MA
c) 4-letter words
i) all letters different
ii) 2 letters same,2 different
iii) 2 pairs same leeters
iv) 3 letters same
Can letters be used twice? Do the words have to be in the dictionary?
no
In that case, for PLANETS, the possible four-letter combinations (in any order) are 5 x 4 = 20. For each combination, there are 4! = 24 ways (permutations) for arranging the letters. That makes 20x24 = 480 possible "words". Most will not be in the dctionary.
Try your own reasoning with the MAMMALS problem. We will be glad to critique your work.
To find the number of 4-letter words that must include both the letter L and A, we can use the concept of permutations. Since there are only two specific letters that need to be included, we can treat them as fixed positions in the word.
Step 1: Identify the fixed positions for the letters L and A.
In a 4-letter word, the possible positions for the letter L and A are as follows:
_L_A_
_L__A
A_L__
A__L_
Step 2: Calculate the number of options for the remaining two letters.
Since there are 26 letters in the English alphabet (ignoring case distinction), there are 26 options for each remaining position.
Step 3: Multiply the number of options for each position.
As there are multiple possibilities for each fixed position, we need to multiply the number of options for each position together.
Total number of 4-letter words with both L and A = 2 (options for L and A) * 26 (options for the remaining two letters) * 26 (options for the remaining two letters) = 2 * 26 * 26 = 1,352
Therefore, there are 1,352 4-letter words that must include both the letter L and A.
Now, let's move on to counting the number of words for the word MAMMALS.
a) To find the number of 7-letter words, we can use the same approach as for the previous question.
Total number of 7-letter words = 7! (7 factorial) = 7 * 6 * 5 * 4 * 3 * 2 * 1 = 5,040
b) To find the number of 7-letter words beginning with MA, we can consider the position of the fixed letters as follows:
MA_____ (remaining 5 positions)
Similar to the previous question, we can calculate the number of options for each remaining position.
Total number of 7-letter words beginning with MA = 26 (options for M) * 26 (options for A) * 26 (options for each of the remaining 5 positions) = 26 * 26 * 26 * 26 * 26 * 26 = 26^5 = 11,881,376
c) For 4-letter words, we need to consider different scenarios based on the given conditions:
i) All letters are different:
Here, we have 26 options for the first letter, 25 options for the second letter (as it cannot be the same as the first), 24 options for the third letter, and 23 options for the fourth letter.
Total number of 4-letter words with all different letters = 26 * 25 * 24 * 23 = 35,880
ii) 2 letters are the same, 2 letters are different:
To count the number of possibilities, we can consider the arrangement of the letters:
XXY_
Here, the first two positions represent the same letter, the third position represents a different letter, and the fourth position can be any letter from the 26 options available.
Total number of 4-letter words with 2 letters the same and 2 letters different = 26 (options for the first letter) * 25 (options for the second letter, as it cannot be the same as the first) * 24 (options for the third letter) * 26 (options for the fourth letter) = 40,560
iii) 2 pairs of the same letters:
Here, we need to consider the arrangement of the letters:
XYYX
The first and fourth positions represent the same letter, while the second and third positions represent another letter.
Total number of 4-letter words with 2 pairs of the same letters = 26 (options for the first letter) * 25 (options for the second letter) * 1 (only one option for the third letter, as it is the same as the second) * 26 (options for the fourth letter) = 16,900
iv) 3 letters are the same:
This scenario includes words such as AAAA, BBBB, CCCC, etc.
Total number of 4-letter words with 3 letters the same = 26 (options for the first letter) * 1 (only one option for the second, third, and fourth letters) = 26
Therefore, for the word MAMMALS:
a) There are 5,040 7-letter words.
b) There are 11,881,376 7-letter words beginning with MA.
c)
i) There are 35,880 4-letter words with all different letters.
ii) There are 40,560 4-letter words with 2 letters the same and 2 letters different.
iii) There are 16,900 4-letter words with 2 pairs of the same letters.
iv) There are 26 4-letter words with 3 letters the same.