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Fill in the Blanks: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

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Q.1. In a certain test, ai students gave wrong answers to atleast i questions, where i = 1, 2, …, k. No student gave more than k wrong answers. The  total number  of wrong answers given is ................. (1982 - 2 Marks)

Ans. Sol.  Number of students who gave wrong answers to exactly one question = a1 – a2, Two questions = a2 – a3
Three questions = a3– a4, k–1 question = ak–1–ak , k question = ak
∴ Total number of wrong answers
=  1  (a1– a2) + 2 (a2 – a3) + 3 (a3 – a4) + .... (k – 1) (ak–1– ak) + k ak
= a+ a+ a3 + ....ak

 

Q.2. The side AB, BC and CA of a triangle ABC have 3, 4 and 5 interior points respectively on them. The number of triangles that can be constructed using  these  interior  points as vertices is ................. (1984 - 2 Marks) 

Ans. 205

Sol. We have total 3 + 4 + 5 = 12 points out of which 3 fall on one line, 4 on other line and 5 on still other line. So number of D‘s that can be formed using 12 such points are =   12C33C4C35C3

Fill in the Blanks: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE = 220 – 15 = 205

 

Q.3. Total number of ways in which six ‘+’ and four ‘–’ signs can be arranged in a line such that no two '–' signs occur together is ................. (1988 - 2 Marks) 

Ans. 35

Sol. ‘+’ signs can be put in a row in 1 way, creating 7 ticked places to keep ‘–’ sign so that no two  ‘–’ signs occur together
√ +√ + √ +√ +√ +√ +√
Out of these 7 places 4 can be chosen in 7C4 ways.
∴ Required no. of arrangements are
= 7C4 = 7C3 = Fill in the Blanks: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

 

Q.4. There are four balls of different colours and four boxes of colours, same as those of the balls. The number of ways in which the balls, one each in a box, could be placed such that a ball does not go to a box of its own colour is _________.

Ans. 9

Sol.  KEY CONCEPT : We know that number of dearrangements of n objects

Fill in the Blanks: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

∴ No. of ways of putting all the 4 balls into boxes of different colour

Fill in the Blanks: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEFill in the Blanks: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Fill in the Blanks: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

The document Fill in the Blanks: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE is a part of the JEE Course Maths 35 Years JEE Main & Advanced Past year Papers.
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