Question

A) How many different strings can be made from the word PEPPERCORN when:______

i) all the letters are used?
ii) at least 6 of the letters are used?
b) How many different strings can be made from the letters in AARDVARK, using all of the letters, if all three As must be consecutive?
c) How many permuations of the 26 letters of the English alphabet do not contain any of the strings fish, rat, or bird?

Answer 1

Part a):

Number of letters = 10

Repeated letters: P=3 times , E = 2 times, R = 2 times

Given word can make strings = 10!/ 3! * 2! * 2! * 1! * 1! * 1!

Total Permutations = 3628800/24 = 151200

Part b):

Given word = AARDVARK

Taking A's as a single letter = AAA RDVRK

Number of strings we can make = 6! / 2! = 360

Part c):

Given that:

Total possible permutations = 26!

Let the permutation for fish = X, rat = Y, bird = Z

X = 23! , Y = 24! , Z = 23!

Now according to given scenario:

|X U Y U Z | = |X| + |Y| + |Z| − |X ∩ Y| − |X ∩ Z| − |Y ∩ Z| + |X∩ Y∩ Z |

As we know:

X intersection Y = Y intersection Z = X intersection Y intersection Z = phi

So, | X U Y U Z | = 24! + 2 · 23! − 21!.

=> 26! − 24! − 2 · 23! + 21!

is the required answer.

i hope it will help you!