Directions: Study the following set of questions carefully and answer the questions given below.
1
In a language, there are six different words. A sentence can be formed by at least 2 words. If order of words is changed in a sentence, we get a different sentence. How many different sentences can be formed in this language?
» Explain it
C
Here, different order gives different sentence. So, permutations are needed to make sentences.
 
⇒ Different sentences that can be formed = 6P2 + 6P3 + 6P4 + 6P5 + 6P6 = 30 + 120 + 360 + 720 + 720 = 1950

Hence, option (C) is correct.
 
2
How many 3 - letter words with or without meaning, can be formed out of the letters of the word, 'LOGARITHMS', if repetition of letters is not allowed?
» Explain it
A
The word 'LOGARITHMS'' has 10 different alphabets

Hence, the number of 3-letter words (with or without meaning) formed by using these letters = (10P3)

= 10 × 9 × 8 = 720

Therefore, option (A) is correct.

3
There are 2 shirts, 3 jeans, 3 socks and 2 skirts. In how manys ways a shopkeeper can arrange these things so that all the socks come together and all the skirts come together?
» Explain it
B
We will count 3 socks as 1 socks and 2 skirts as 1 skirt.

Total ways = 7!

= 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040

Hence, option B is correct.
4
In how many different ways can the letters of the word WINDOW be arranged in such a way that the vowels never come together?
» Explain it
D
Total ways = 6! / 2! = 360

ways when all the vowels always come together = 5! / 2! = 60

Ways when all the vowels never come together = 360 – 60 = 300

Hence, option D is correct.
5
The number of ways in which 8 different books can be arranged on a shelf so that 3 particular books shall not be together:
» Explain it
D
Number of ways in which 8 books can be arranged = 8!

Number of ways when three particular books are together = 6! X 3!

Therefore Number of ways when three particular books are not together = 8! - 6! X 3!

= 6!(7 x 8 - 3 x 2)

= 6! X 50 = 720 x 50 = 36000

Hence, option D is correct.