How many different letter arrangements can be made from the letters (a) Fluke? (b) Propose? (c) Mississippi? (d) Arrange?

We have to find the different letter arrangements for Fluke

In the word Fluke there are 5 different letters, then the different letter arrangement for ' the word Fluke is$$\begin{gathered} 5!=5\times 4\times 3\times 2\times 1 \\ =120 \end{gathered}$$

we have to find different letter arrangements for Propose

here are 7 seven letters in the word Propose, out of letters 2 identical letters of p and o. So to get different letter arrangements for the given word we have to divide 7! by identical word arrangements 2! and 2!. Then the different letter arrangements for the word Propose is

$$\begin{gathered} \frac{7!}{2!2!}=\frac{7\times 6\times 5\times 4\times 3\times 2\times 1}{2\times 1\times 2\times \text{'}1} \\ =1260 \end{gathered}$$

Here we have to find different letter arrangements for Mississippi

In the word Mississippi, there are a total of 11 letters out of 11, there are d identical i letter, 4 identical s letters, and 2 identical letters p. Then to get a different word arrangement we have to divide 11! by each identical letter arrangement, thus the different letter arrangements for the word is

$$\begin{gathered} \frac{11!}{4!\times 4!\times 2!}=\frac{11\times 10\times 9\times 8\times 7\times 6\times 5\times 4!}{4\times 3\times 2\times 1\times 4!\times 2\times 1} \\ =34650 \end{gathered}$$

we need to find different letter arrangements for Arrange

in this word Arrange there are a total of 7letters, in which 2 identical letters a and 2 identical letters r are there. To get different letter arrangements we have to divide 7! by identical arrangements, then the arrangements will be

$$\begin{gathered} \frac{7!}{2!\times 2!}=\frac{7\times 6\times 5\times 4\times 3\times 2!}{1\times 2\times 2!} \\ =1260 \end{gathered}$$