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There are 40 lines in a plane of which a set of 12 lines are concurrent at A, another set of
15 lines are concurrent at B and the set of remaining lines are parallel. What is the number
of points of intersection of these 40 lines, given that the three sets are disjoint?
(A) 611 (B) 531 (C) 533 (D) 638

 

 

TG Team 30/10/2018 8:00 pm

Total number of  points of intersection of 40 points  :   40C2 = 780 

 

Number of points of intersection when lines are parallel : 0 

So remove 13C2 i.e 78 cases 

12 lines are concurrent at A so remove 12C2 i.e 66 cases 

15 lines are concurrent at B so remove 15C2 i.e 105 points 

 

Hence , 780 - 105 - 78 - 66 + 2 = 533 intersection points . 

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There are 33 similar chocolates to be distributed among 7 children such that each child gets distinct and natural number of chocolates. In how many ways it is possible ? 

TG Team 14/11/2018 11:43 pm

first distribute 1, 2, 3,...., 7 chocolates 

Remaining : 38 - 28 = 5 chocolates 

now 5 chocalates can be distributed in

1 , 1, 1, 1, 1
1 , 1,1 2 
1 , 1, 3 
1, 4 
1, 2, 2 
2,3 

7 ways 

Further these 7 children can be arranged in 7! ways 

Hence , 7 x 7! ways. 

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Sir , can you share the solution to this problem? I am getting B part as my answer , but in the key the ans given is C 

IMG 20181115 095545

 

This post was modified 6 years ago by Nikita Chawla
TG Team 15/11/2018 11:55 am

Hi Nikita 

B is the correct answer. 

Total number of triangles formed is nC3 , now consider the following cases –

1. Number of triangles having three sides common with the sides of polygon is 0.
2. Number of triangles having two sides common with the sides of polygon is n.
3. Number of triangles having one side common with the sides of the polygon = number of ways of selecting 3 points (vertices )  out of which only two are consecutive is n(n-4)C1 = n(n-4)
So number of required triangles is nC3 – n – n (n-4) = n ( n-4) (n-5)/6

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81 students appeared the examinations in physics, chemistry and biology. 46 students passed in physics, 47 students passed in chemistry, and 48 students passed in biology; 29 students passed in physics and chemistry, 27 students passed in chemistry and biology, and 13 students passed in physics and biology. The largest possible number that could have passed all the three examinations is:
TG.Raman 16/11/2018 12:16 pm

E1+2E2+3E3 = 46+47+48

E1+E2+E3 = 81 (consider none fails for maximum value)

E2+3E3 = 29+ 27+ 13

Solve for E3 to get 9

Here E1 = number of students passed in exactly one exam 
E2 = passed in exactly two exams 
E3 = Number of students passed in all three exams. 

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Is there any shortcut to find the number of squares in a rectangular chessboard? (eg. Number of squares in 8X10 chessboard?)

TG.Raman 17/11/2018 3:25 pm

Number_of_squares_and_rectangles_in a m x n matrix

Case 1 : m = n = k(say)

Squares = 1² + 2²+ .. + k² = k(k + 1)(2k + 1)/6

Rectangles = 1³+ 2³ + .. + k³ = [k(k + 1)/2]²

Case 2 : m ≠n
Squares = mn + (m - 1)(n - 1) + (m - 2)(n - 2) + .. + 0

Rectangles = (1 + 2 + 3 + .. + m)(1 + 2 + .. + n) = m(m + 1)n(n + 1)/4

TG.Raman 17/11/2018 3:28 pm

Hence, the number of squares in a 8 x 10 rectangular grid : 

10 x 8 + 9 x 7 + 8 x 6 + 7 x 5 + 6 x 4 + 5 x 3 + 4 x 2 + 3 x 1 = 276 squares

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15425307083392122917219
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3 points are randomly selected on a sphere,what is the probability that all of them lie in the same hemisphere?

 

Should it not be?-2πrsq./4πrsq.

TG Team 18/11/2018 11:40 pm

No! 

It is 1. 100% probable. 

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1542531247661421879991

Sir,kindly post the Solution of  both the questions

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An institute has 5 departments and each department has 50 students. If students are picked up randomly from all 5 departments to form a committee, what should be the minimum number of students in the committee so that at least one department should have representation of minimum 5 students?

1. 11

2. 15

3. 21

4. 41

5. None of the above

TG Team 14/12/2018 7:09 pm

The maximum number of students can be picked from each department such that 5 students are not selected from the same department is 4.

Therefore, after 4 students from each department are selected i.e 4 x 5 = 20 ,  the 21st student selected will be the fifth student to be selected from one of the 5 departments.

Hence, , 20+1 = 21 students should be selected in total to ensure that at least five students from one of the departments is selected. 

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In how many ways can we select 3 numbers from first 20 positive integers, such that their product is always divisible by 5?

Sir, won't the answer be 580?

TG.Raman 15/12/2018 1:02 pm

yes 580 is the correct answer . 
Approach : 1 
select 3 numbers in 20C3 ways. 

cases when none of the selected number is a multiple of 5 : 16C3 ways 

20C3 - 16C3 = 580 

Approach : 2 
4C1 x 16C2  + 4C2 x 16C1 + 4C3 = 580

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How many 3-letter words can be formed using the letters of the word 'INSTITUTE'?

Gaurav Jain 12/07/2019 3:40 pm

case 1 - when all letters are distinct
then there are 6 different letters hence 6P3 = 120

Case 2 - when 1 letter repeated twice
then it can be either I or T means 2C1 cases and 1 more letter means 4C1 case
further arrangements 3! , hence total 2C1*4C1*3! = 48

Case 3 - when all letters are same , that is only one case TTT

hence total 169 cases

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In how many ways can you select exactly 7 letters from 3A, 4B, 2C and 1 D?

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x+ y+ z=19, x≤ y≤ z, find total number of whole number solutions?

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Find the number of ways to distribute 2n different balls into
(a) n equal lots
(b) equally among n children

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Hello, Please solve :

How many different size rectangles are there in a square grid of 5 x 7 ?

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