Thread: Refreshment of permutation and combination

(1) How many four letter words can be formed using the letters of the word EQUATION so that the letter Q is included in each word ?

isn' t the answer 7x6x5 = 210 ... Am i correct ?


(2) How many five letter words can be formed by picking 3 vowels and 2 consonants from the word REACTION such that two consonants always remain together ?

Originally Posted by mathaddict

(1) How many four letter words can be formed using the letters of the word EQUATION so that the letter Q is included in each word ?

isn' t the answer 7x6x5 = 210 ... Am i correct ?

No

If Q is at the first place then you have 7 choices for the second letter then 6 choices for the third one and finally 5 choices for the fourth one
That makes 7x6x5 = 210
But if Q is at the second place you have also 210 possibilities. The same if Q is at the third place and the same for the fourth place.
Therefore there are 210 x 4 = 840 words

Another way is to calculate the number of group of 4 letters containing Q
There are groups of 4 letters with Q (the same number as the number of groups of 3 letters among 7 letters)
For each group of 4 letters you can build 4! = 24 words
The total number of words is 35 x 24 = 840

Hello, mathaddict!

Another approach . . .

(1) How many 4-letter words can be formed using the letters of the word EQUATION
so that the letter Q is included in each word?

The letter Q will be included.
Select 3 letters from the other 7 letters: . ways.

We have a set of 4 letters; they can be arranged in: ways.

Therefore, there are: . possible words.

(2) How many 5-letter words can be formed by picking 3 vowels and 2 consonants
from the word REACTION such that two consonants always remain together?

There are 4 vowels and 4 consonants.

Select 3 of the 4 vowels: . ways.
Select 2 of the 4 consonants: . ways.

. . We have: . selections of letters.


We have 3 vowels and 2 consonants: .

Duct-tape the two consonants together: .

These four "letters" can be arranged in: ways.
But the two constants could be ordered either or .
. . Hence, there are: . orders.

Therefore, there are: . possible words.

Originally Posted by Soroban

Hello, mathaddict!

Another approach . . .

The letter Q will be included.
Select 3 letters from the other 7 letters: . ways.

We have a set of 4 letters; they can be arranged in: ways.

Therefore, there are: . possible words.

Not so new !
See my post