When all letters are used, how many different ’words’ can be made from the letters
(a) PORPOISE
(b) MANATEE
(c) EFFIGY

To find the number of different "words" that can be made from the letters of each word when all letters are used, we need to calculate the number of distinct permutations of the letters in each word. This involves accounting for repeated letters by dividing the total number of permutations by the factorials of the frequencies of each repeated letter.

### (a) **PORPOISE**
The word "PORPOISE" contains 8 letters, where:
- P occurs 2 times,
- O occurs 2 times,
- R, I, and S each occur 1 time.

The total number of different "words" is calculated using the formula for permutations of multiset:

\[
\frac{8!}{2!2!}
\]

Now, calculate:

\[
8! = 40320, \quad 2! = 2
\]

\[
\frac{40320}{2! \times 2!} = \frac{40320}{2 \times 2} = \frac{40320}{4} = 10080
\]

So, **10,080** different words can be formed from "PORPOISE."

---

### (b) **MANATEE**
The word "MANATEE" contains 7 letters, where:
- A occurs 2 times,
- E occurs 2 times,
- M, N, and T each occur 1 time.

The total number of different "words" is:

\[
\frac{7!}{2!2!}
\]

Now, calculate:

\[
7! = 5040, \quad 2! = 2
\]

\[
\frac{5040}{2! \times 2!} = \frac{5040}{2 \times 2} = \frac{5040}{4} = 1260
\]

So, **1,260** different words can be formed from "MANATEE."

---

### (c) **EFFIGY**
The word "EFFIGY" contains 6 letters, where:
- F occurs 2 times,
- E, I, G, and Y each occur 1 time.

The total number of different "words" is:

\[
\frac{6!}{2!}
\]

Now, calculate:

\[
6! = 720, \quad 2! = 2
\]

\[
\frac{720}{2} = 360
\]

So, **360** different words can be formed from "EFFIGY."

---

### Summary:
- (a) PORPOISE: **10,080** words
- (b) MANATEE: **1,260** words
- (c) EFFIGY: **360** words

---

Do you need more details or have any further questions?

Here are 5 related questions for you to explore:
1. How many words can be made from the letters in "BALLOON"?
2. What if the word "PORPOISE" had no repeated letters, how many distinct words could be made?
3. How many distinct permutations can be made from the word "BANANA"?
4. How would the formula change if all the letters in a word were distinct?
5. How many words can be formed by rearranging the letters of "BUTTERFLY"?

**Tip:** When letters repeat, always divide the total number of permutations by the factorials of the number of times each letter repeats.

When all letters are used, how many different ‘words’ can be made from the letters (a) PORPOISE (b) MANATEE (c) EFFIGY

Find out how many distinct words can be formed by rearranging the letters of 'PORPOISE,' 'MANATEE,' and 'EFFIGY' using permutation formulas. This problem uses key concepts of combinatorics and factorials, making it suitable for students in Grades 9-12.

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