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Number of integral points (integral poin...

Number of integral points (integral points means both the co-ordinates should be integer) exactly in the interior of the triangle with vertices `(0,0),(0, 15) and Or (15, 0)` is

A

133

B

190

C

233

D

105

Text Solution

Verified by Experts

The correct Answer is:
B
Clearly ,

Required number of integral points
= `(1)/(2)` (Number of integral points in the interior of square OACB - Number of integral points on line AB)
Now , Number of integral points on AB
= Number of positive integral solutions of x + y = 21
`= ""^(21 -1)C_(2-1) = ^(20)C_(1) = 20`
Hence , required number of points = `(1)/(2) (20 xx 20 -20) = 190 `
ALTER Coordinates of integral points inside the triangle OAB are :
`(1,1) , (2,1) ..., (1,18) , (1,19)`
(2,1) , (2,2) ..., (2,18)
(3,1) , (3,2) ..., (3,17)
` " " : " " :`
(17 , 1) , (17, 2) , (17 , 3)
(18 , 1) , (18,2)
(19,1)
`therefore` Total number of integral points
`= 19 + 18 + 17 + ... + 2 + 1 = (19 xx20)/(2) = 190`
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