JEE Advanced (Fill in the Blanks): Permutations & Combinations | Chapter-wise Tests for JEE Main & Advanced PDF Download

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Q.1. In a certain test, a_i 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 = a_k–1–a_k , k question = a_k
∴ Total number of wrong answers
=  1  (a₁– a₂) + 2 (a₂ – a₃) + 3 (a₃ – a₄) + .... (k – 1) (a_k–1– a_k) + k a_k
= a₁ + a₂ + a₃ + ....a_k

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 =   ¹²C₃ – ³C₃ – ⁴C₃ – ⁵C₃

 = 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 =

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

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