Question

How many di↵erent strings can be made from the letters in the work PEPPERCORN when all the letters are used? How may of the strings start and end with the letter P? How many strings have 3 consecutive Ps?

Answer 1

420 different strings can be made from the letters in the word PEPPERCORN when all the letters are used. 15120 strings start and end with the letter P. 20160 strings have 3 consecutive P's.

Step-by-step explanation:

To find the number of different strings that can be made from the letters in the word PEPPERCORN when all the letters are used, we need to calculate the permutations of the letters. The word PEPPERCORN has 10 letters, with 2 E's, 3 P's, 1 C, 1 O, 1 R, and 1 N. We can use the formula for permutations with repeated elements to determine the number of different strings. The formula is:

P = n! / (r1! * r2! * ... * rn!)

Where n is the total number of objects and r1, r2, ..., rn are the number of repeated objects. In this case, n = 10, r1 = 2, and r2 = 3. Plugging these values into the formula, we get:

P = 10! / (2! * 3!) = 5040 / (2 * 6) = 420

Therefore, 420 different strings can be made from the letters in the word PEPPERCORN when all the letters are used.

To find the number of strings that start and end with the letter P, we can treat the letter P as a fixed position. We have 9 positions left to fill with the remaining letters, which include 1 E, 2 P's, 1 C, 1 O, 1 R, and 1 N. Using the same formula as above, we get:

P' = 9! / (1! * 2!) = 30240 / 2 = 15120

Therefore, 15120 strings start and end with the letter P.

To find the number of strings that have 3 consecutive P's, we can treat the 3 P's as a single element. We have 8 positions left to fill with the remaining letters, which include 2 E's, 1 C, 1 O, 1 R, and 1 N. Using the formula again, we get:

P'' = 8! / (2!) = 40320 / 2 = 20160

Therefore, 20160 strings have 3 consecutive P's.

Answer 2

a. 151,200 ways b. 10,080 ways c. 10,080 ways

Step-by-step explanation:

a. How many different strings can be made from the letters in the work PEPPERCORN when all the letters are used?

Since there are 10 letters in the word PEPPERCORN, there are 10! ways of arranging them. Since we have 3 P's, there are 3! ways of arranging them . We have 2 E's, there are 2! ways of arranging them. We have 2 R's, there are 2! ways of arranging them. We have 1 O, there are ! ways of arranging them. We have 1 C, there are 1! ways of arranging them. We have 1 N, there are 1! ways of arranging them. So there are

`(10!)/(3!2!2!1!1!1!) = 151,200 ways` of arranging the word

b. How may of the strings start and end with the letter P?

If the strings start and end with P, then we have 8 letters left including 1 P.

Since there are 8 letters left in the word PEPPERCORN, there are 8! ways of arranging them. Since we have 1 P, there are 1! ways of arranging them . We have 2 E's, there are 2! ways of arranging them. We have 2 R's, there are 2! ways of arranging them. We have 1 O, there are ! ways of arranging them. We have 1 C, there are 1! ways of arranging them. We have 1 N, there are 1! ways of arranging them. So there are

`(8!)/(2!2!1!1!1!1!) = 10,080 ways` of arranging the word

c. How many strings have 3 consecutive Ps?

There are 8 different ways of arranging the P's consecutively. We are then left with 7 letters. Since there are 7 letters left in the word PEPPERCORN, there are 7! ways of arranging them. We have 2 E's, there are 2! ways of arranging them. We have 2 R's, there are 2! ways of arranging them. We have 1 O, there are ! ways of arranging them. We have 1 C, there are 1! ways of arranging them. We have 1 N, there are 1! ways of arranging them. So there are

`(8!)/(2!2!1!1!1!1!) = 10,080 ways` of arranging the word