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IIT JEE Main Maths -Unit 4 -Combinations- Exam Style Questions- New Syllabus

IIT JEE Main Maths - Exam Style Practice Questions - All Chapters
Question 
From all the English alphabets, five letters are chosen and arranged in alphabetical order. The total number of ways in which the middle letter is ‘M’ is:
\((1)\ 14950\)
\((2)\ 6084\)
\((3)\ 4356\)
\((4)\ 5148\)
▶️ Answer/Explanation
There are 12 letters before M and 13 after. We must choose 2 from before and 2 from after: \(\binom{12}{2} \times \binom{13}{2} = 66 \times 78 = 5148\).
The correct answer is \((4)\ 5148\).
Question 
In a group of 3 girls and 4 boys, there are two boys $B_1$ and $B_2$. The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but $B_1$ and $B_2$ are not adjacent to each other, is :
(1) 144
(2) 72
(3) 96
(4) 120
▶ Answer/Explanation
Ans. (1)
Sol. Total – when $B_1$ and $B_2$ are together
= 2!(3! 4!) – 2! (3!(3! 2!)) = 144
Question 

The number of words, which can be formed using all the letters of the word “DAUGHTER”, so that all the vowels never come together, is

(1) 34000
(2) 37000
(3) 36000
(4) 35000

▶️ Answer/Explanation

Ans. (3)

Sol. DAUGHTER
Total words = \(8!\)
Total words in which vowels are together = \(6! \times 3!\)
words in which all vowels are not together
\(= 8! – 6! \times 3!\)
\(= 6! [56 – 6]\)
\(= 720 \times 50\)
\(= 36000\)
Ans.(3)

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