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27 Nov 2024, 01:30
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C
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Difficulty:
35%
(medium)
Question Stats:
69% (01:29) correct
31%
(01:29)
wrong
based on 166
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If 6 points are indicated on a line and 8 points are indicated on another line that is parallel to the first one, how many quadrilaterals can be formed whose vertices are among the 14 points?
A. 42
B. 43
C. 420
D. 1001
E. 1680
A. 42
B. 43
C. 420
D. 1001
E. 1680
27 Nov 2024, 05:06
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IMO Ans: C
To form a quadrilateral, we need to select 2 points from a line with 6 points and another 2 points from a line with 8 points.
thus, the number quadrilaterals that can be formed is:
6C2 x 8C2 = 15 x 28 = 420
To form a quadrilateral, we need to select 2 points from a line with 6 points and another 2 points from a line with 8 points.
thus, the number quadrilaterals that can be formed is:
6C2 x 8C2 = 15 x 28 = 420
27 Nov 2024, 23:57
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-------A1-------A2-------A3-------A4--------A5-------A6--------
------------B1------------B2-------B3--------B4-------B5---------B6-------B7-----B8----------
Consider the above two parallel lines with 6 points (A1 to A6) and 8 points (B1 to B8) respectively.
Now, a quadrilateral consists of four vertices. Accordingly, we have to select four points from the two parallel lines to form a quadrilateral. There are only following three ways to select four points on the two lines:-
Case I - All four vertices on the same line
Joining the four vertices will give us a line, not a quadrilateral.
Case II - Three vertices on one line and one vertex on other line
This case will give us a triangle, not a quadrilateral.
Case III - Two vertices on one line and two vertices on other line.
This will give us a quadrilateral.
Now, it is important to understand that for any four points selected as per Case III mentioned above, we will get only one quadrilateral. Ex-if you select A1 and A3 on first line and B2 and B4 on second line, you can only form one quadrilateral using A1, A3, A2 and B4. Hence, for any combination of four points on two lines such that two points lie on one line and two points lie on other line, we will get one quadrilateral. Accordingly, the number of quadrilaterals which we can make is same as number of ways to choose four points on two lines such that two points lie on one line and two points lie on other line.
Hence, the number of quadrilaterals = 6C2 * 8C2 = 15 * 28 = 420. OPTION (C).
------------B1------------B2-------B3--------B4-------B5---------B6-------B7-----B8----------
Consider the above two parallel lines with 6 points (A1 to A6) and 8 points (B1 to B8) respectively.
Now, a quadrilateral consists of four vertices. Accordingly, we have to select four points from the two parallel lines to form a quadrilateral. There are only following three ways to select four points on the two lines:-
Case I - All four vertices on the same line
Joining the four vertices will give us a line, not a quadrilateral.
Case II - Three vertices on one line and one vertex on other line
This case will give us a triangle, not a quadrilateral.
Case III - Two vertices on one line and two vertices on other line.
This will give us a quadrilateral.
Now, it is important to understand that for any four points selected as per Case III mentioned above, we will get only one quadrilateral. Ex-if you select A1 and A3 on first line and B2 and B4 on second line, you can only form one quadrilateral using A1, A3, A2 and B4. Hence, for any combination of four points on two lines such that two points lie on one line and two points lie on other line, we will get one quadrilateral. Accordingly, the number of quadrilaterals which we can make is same as number of ways to choose four points on two lines such that two points lie on one line and two points lie on other line.
Hence, the number of quadrilaterals = 6C2 * 8C2 = 15 * 28 = 420. OPTION (C).
Bunuel
