Question

a) How many different strings can be made from the word PEPPERCORN when (SHOW WORK & Explaination)

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? (SHOW WORK & Explaination)

c) How many permuations of the 26 letters of the English alphabet do not contain any of the strings fish, rat, or bird? (SHOW WORK & Explaination)

Answer 1

Answer :

Total letters = 10

Letters repeated = P = 3 times

E = 2

Therefore , strings can be made from the given word = 10 ! / 3! x 2 ! x 1 ! x 1 ! x 1!

b) Let us treat the consectives A's as one letter : AAA RDVRK

Therefore different strings can be made = 6! / 2! = 360

c) Total permutations possible = 26 !

Let X be the set of permutations which contains fish , Y and Z similarly for rat and bird

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

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

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

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

=> 26! − 24! − 2 · 23! + 21!