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 C36
Considering case (i)
Since, a,b,c,d,e are the angle of a convex pentagon
so, a+b+c+d+e=360o ...(i)
Now, Given that a≤b≤c≤d≤e
and since they are angles of polygon
So a>0o ...(ii)
Now, consider the point (iii), they are in A.P.
∴ Let a1 be the first angle and r be the common difference
Then a,b,c,d,e can be assumed as,
a1−2r,a1−r,a1,a1+r,a1+2r
Now, substituting these values in (i)
a1−2r+a1−r+a1+a1+a1+r+a1+2r=360o
or, a1=72o
Now, d can take any values but considering the facts (ii)