Question

Find the number of five letter words that can be formed from the letters of the word PROBLEMS. ( please explain with details.)

(a) How many of them contain only consonants?

(b) How many of them begin and end in a consonant?

(c) How many of them begin with a vowel?

(d) How many contain the letter S?

(e) How many begin with B and also contain S?

(f) How many begin with B and end in a vowel?

(g) How many contain both vowels?

Answer 1

Please note nCr = n!/ [(n-r)! * r!]

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There are 2 Vowels (O and E) and the remaining 6 letters are consonants (P, R, B, L, M and S). Total 8 letters.

Assuming that repetition is not allowed, as nothing is mentioned

(a) How many 5 letter words contain only consonants?

consonant	consonant	consonant	consonant	consonant
6	5	4	3	2

In the first box, any of the 6 consonants can go in or be chosen. The second box can take any of the remaining 5, the 3rd box any remaining 4, the 4th box any remaining 3 and the last box can take any of the remaining 2 consonants.

Therefore total possible cases = 6 * 5 * 4 * 3 * 2 = 720

We can also do it in the following manner: choose any 5 consonants in 6C5 ways = 6, then arrange these 5 chosen in 5! ways = 120 ways. Therefore total ways = 6 * 120 = 720.

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(b) How many 5 letter words begin and end with consonants

The fir box can take of the 6 consonants, and the last box can take any of the 5 remaining consonants.

consonant	C/V	C/V	C/V	consonant
6	6	5	4	5

There are no restrictions for the 2nd 3rd and 4th box, They can take vowels or consonants, There are 6 such letters remaining (from 8, the first and last are gone). The second place can take any of the 6 letters, the third place any 5, and the 4th place any 4.

Therefore total possibilities = 6 * 6 * 5 * 4 * 5 = 3600

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(c) How many 5 letter words are there that begin with a vowel?

The first place can take any of the 2 vowels in 2C1 = 2 ways. There are 7 letters remaining. The second place can take any of the 7 letters, the third place any 6, and the 4th place any 5 and the 5th place any 4.

Vowel	V/C	V/C	V/C	V/C
2	7	6	5	4

Therefore total possibilities = 2 * 7 * 6 * 5 * 4 = 1680

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(d) How many 5 letter words are there which contain S

= Total Number of words - Those words which do not have S,

The total 5 letter words possible:

The first place can take any of the 8 letters, the second place can take any of the 7 letters, the third place any 6, and the 4th place any 5 and the 5th place any 4.

V/C	V/C	V/C	V/C	V/C
8	7	6	5	4

Therefore total possibilities = 8 * 7 * 6 * 5 * 4 = 6720

Now, Number of words which don't have S.

The first place can take any of the 7 letters, the second place can take any of the 6 letters, the third place any 5, and the 4th place any 4 and the 5th place any 3.

Therefore total possibilities without S = 7 * 6 * 5 * 4 * 3 = 2520

Therefore the number of words that have an S = 6720 - 2520 = 4200

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(e) How many 5 letter words begin with B and contain S?

= All the letters that begin with B - The words that begin with B and don't contain S.

Total Number of words that begin with B

Fixed	V/C	V/C	V/C	V/C
B	7	6	5	4

The first place can take only B, the second place can take any of the 7 letters, the third place any 6, and the 4th place any 5 and the 5th place any 4.

Therefore total possibilities = 1 * 7 * 6 * 5 * 4 = 840

Now Total words starting with B and without S :

The first place can take only B, the second place can take any of the 6 letters, the third place any 5, and the 4th place any 4 and the 5th place any 3 = .1 * 6 * 5 * 4 * 3 = 360

Therefore cases starting with B and having S = 840 - 360 = 480

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(f) How many 5 letter words are there beginning with B and ending in a Vowel

Here the first place is fixed with B, and the 5th place can take any of the 2 vowels. There are 6 letters remaining. The second place takes any 6, the 3rd place any 5 and the 4th place any of the remaining 4 letters.

Therefore Total number of ways = 1 * 6 * 5 * 4 * 2 = 240

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(g) How many 5letter words can be formed that contain both vowels?

= Total number of words - Words that contain 0 vowels - words that contain 1 vowel

From (d) we know that total possible words = 6720

Words that contain 0 vowels (all consonants). From (a), this value = 720

Words that contain 1 vowel and the remaining 4 are consonants: choose 1 vowel out of 2 in 2C1 = 2 ways and choose 4 consonants out of 6 in 6C4 ways and then arrange these 5 chosen letters in 5! = 120 ways.

The total such cases containing exactly 1 vowel = 2 * 15 * 120 = 3600

Therefore the cases where we have both vowels = 6720 - 720 - 3600 = 2400

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