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Question

# The number of 5-tuples (a, b, c, d, e) of positive integers such that.I. a, b, c, d, e are the measures of angles of a convex pentagon in degrees;II. a≤b≤c≤d≤e.III. a, b, c, d, e are in arithmetic progression is?

A
35
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B
36
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C
37
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D
126
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Solution

## The correct option is C 36Considering case (i)Since, a,b,c,d,e are the angle of a convex pentagonso, a+b+c+d+e=360o ...(i)Now, Given that a≤b≤c≤d≤eand since they are angles of polygonSo a>0o ...(ii)Now, consider the point (iii), they are in A.P.∴ Let a1 be the first angle and r be the common differenceThen a,b,c,d,e can be assumed as,a1−2r,a1−r,a1,a1+r,a1+2rNow, substituting these values in (i)a1−2r+a1−r+a1+a1+a1+r+a1+2r=360oor, a1=72oNow, d can take any values but considering the facts (ii)a>0oor a1−2r>0oor, a1>2ror, r<a12or, r<72o2=36o∴r can lie from 0,1,2.....35∴ No. of tuples =36

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