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 letters
iv) 3 letters same

I am not doing the problem for you but perhaps I can give you an idea.

Let's see. I have to work with
M M M A A L S
If all letters were different the answer would be 7!, but they are not.

Let me try a simpler example
Say
A A B B
A B A B
A B B A
B A A B
B B A A
B A B A Hmm six of them, not 4!
Looks like permutations of 4 items divided by permutations of 2 As divided by permutations of 2 Bs
4!/(2!*2!) = 4*3*2 /( 2*2) = 6
Let's try it with 3 As and 1 B
A A A B
A A B A
A B A A
B A A A Is that 4!/(3!*1!) ?
4*3*2 /(3*2*1) = 4, sure enough.

a) 7-letter words for mammals:

To find the number of 7-letter words for mammals, we need to know the total number of mammals species. However, since there are thousands of mammals species, it's not possible to provide an exact number.

b) 7-letter word beginning with "MA" for mammals:
Similarly, without a specific list of mammals, it's difficult to determine the number of 7-letter words beginning with "MA".

c) 4-letter words for mammals:
i) All letters different: To find all the 4-letter words for mammals with all different letters, we can use combinatorics. There are 26 letters in the English alphabet, so we have 26 options for the first letter, 25 options for the second letter, 24 options for the third letter, and 23 options for the fourth letter. Therefore, the number of 4-letter words with all different letters is: 26 x 25 x 24 x 23 = 35,880 words.

ii) 2 letters same, 2 different: In this case, we can choose the two different letters from the 26 available options in 26C2 ways. Since the two similar letters can be placed in any of the 4 positions, we multiply the number obtained by 4, resulting in: 4 x 26C2 = 4 x 325 = 1,300 words.

iii) 2 pairs same letters: For this scenario, we have 26 options for the first pair of letters and 25 options for the second pair. Additionally, we can arrange the pairs in two ways (ABCD or CDAB). Therefore, the number of 4-letter words with two pairs of the same letters is: 26 x 25 x 2 = 1,300 words.

iv) 3 letters same: In this case, we have 26 options for the first letter, and for the remaining three positions, we need to choose one letter, resulting in 26^1 = 26 possibilities. Therefore, the number of 4-letter words with three of the same letters is: 26 x 26 = 676 words.

To find the number of words that meet the given criteria, we can break down the problem step by step. Let's start with finding the number of 7-letter words for mammals.

a) 7-letter words:
To find the number of 7-letter words, you need to consider all possible combinations of letters. In this case, since we are looking at the word length of 7, we have 7 positions to fill with letters. For each position, we have 26 choices (the alphabet).

So, to find the number of 7-letter words for mammals, we use the formula: 26^7. This is because each position has 26 choices, and there are 7 positions.

b) 7-letter words beginning with "MA":
To find the number of 7-letter words that begin with "MA," we need to fix the first two positions as "MA" and then fill the remaining 5 positions with any letter (26 choices).

So, for this case, we have "MA" as the fixed prefix followed by 26 choices for the remaining 5 positions. To find the number of words, we multiply the number of choices for each position: 26^5.

c) 4-letter words:
Now let's move on to 4-letter words, which have additional criteria. We need to consider different scenarios based on the given conditions.

i) All letters different:
In this case, we can choose any 4 different letters from the alphabet. We have 26 choices for the first position, 25 choices for the second position (excluding the previously chosen letter), 24 choices for the third position, and finally 23 choices for the fourth position.

To find the number of 4-letter words with all different letters, we multiply the number of choices for each position: 26 * 25 * 24 * 23.

ii) 2 letters same, 2 different:
For this scenario, we need to choose 2 different letters from the alphabet and then choose 2 positions to place them. We have 26 choices for the first letter, 25 choices for the second letter, 4 choices for choosing the 2 positions for the same letters (combination of selecting 2 positions from 4 available positions: 4C2), and 3 choices for choosing the positions for the different letters (remaining 2 positions).

To find the number of 4-letter words with 2 letters the same and 2 different letters, we multiply the number of choices for each step: 26 * 25 * 4C2 * 3.

iii) 2 pairs of same letters:
In this case, we need to choose 2 different letters to create the pairs and then choose the 2 positions for each pair. We have 26 choices for the first letter, 25 choices for the second letter, 4 choices for choosing the 2 positions for the first pair (combination of selecting 2 positions from 4 available positions: 4C2), and 2 choices for choosing the 2 positions for the second pair (remaining 2 positions).

To find the number of 4-letter words with 2 pairs of the same letters, we multiply the number of choices for each step: 26 * 25 * 4C2 * 2C2.

iv) 3 letters the same:
In this case, we need to choose 1 letter to repeat three times and then choose the position for the remaining letter. We have 26 choices for selecting the repeating letter and 4 choices for choosing the position for the remaining letter (out of the 4 available positions).

To find the number of 4-letter words with 3 letters the same, we multiply the number of choices for each step: 26 * 4.

By following these steps, we can determine the number of words that match the given criteria for mammals.