11.
How many diagonals are there in a polygon of n sides?
• A.
n (n -3)
•  C. 1 n (n -3) 2
•  B. 1 n (n -3) 3
•  D. 1 n (n -3) 2
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Explanation :
A polygon of n sides has n vertices.
By joining any two of these vertices, we obtain either a side or a diagonal of the polygon.
Number of all straight lines = nC2 =
 n(n-1) 2

Number of diagonals =
 n(n-1) 2
- n =
 1 2
n (n - 3)
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12.
The number of ways in which 4 distinct balls can be put in 4 boxes labeled A, B, C, D, so that one box remains empty is
• A.
232
• C.
196
• B.
192
• D.
144
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Explanation :
 Let the balls be named a,b,c,d and the four boxes be A,B,C and D. One box should always remain empty. Let us assume initially, that the box A remains empty. Now there are 4 balls to be put in 3 Boxes B, C, D. Hence one of the 3 boxes should receive 2 balls. Let this box be B. The two balls to be put in B can be chosen in 4C2 = 6 ways. The box C can be filled with 1 ball out of 2 balls in 2C1 = 2 ways. Hence if A is kept empty and B is filled with 2 balls and C and D are filled with 1 ball in 4C2 x 2 = 12 ways. Similarly, we can put 2 balls in box C or 2 balls in box D, Keeping A empty. Hence it can be done in 12 x 3 = 36 ways. Since any of the 4 boxes, A, B, C and D could be empty we have in all 36 x 4 = 144 ways.
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13.
There are P points in a plane of which only K points are in a straight line. Find the umber of triangles which can be formed by the P points?
•  A. 1 [P(P-1)(P-2) - K(K - 1)(K - 2)] 6
• C.
PC3 - K
• B.
P! - K!
• D.
P! - KC3
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