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. $a\le b\le c\le d\le e$.
III. a, b, c, d, e are in arithmetic progression is?

• A. $35$
• B. $36$
• C. $37$
• D. $126$

Answer 1

Answer: (B)

$36$

Considering case (i)

Since, $a,b,c,d,e$ are the angle of a convex pentagon

so, $a+b+c+d+e={360}^o$ ...(i)

Now, Given that $a\le b\le c\le d\le e$

and since they are angles of polygon

So $a>0^o$ ...(ii)

Now, consider the point (iii), they are in A.P.

$\therefore$ Let $a_1$ be the first angle and $r$ be the common difference

Then $a,b,c,d,e$ can be assumed as,

$a_1-2r,a_1-r,a_1,a_1+r,a_1+2r$

Now, substituting these values in (i)

$a_1-2r+a_1-r+a_1+a_1+a_1+r+a_1+2r={360}^o$

or, $a_1={72}^o$

Now, $d$ can take any values but considering the facts (ii)

$a>0^o$

or $a_1-2r>0^o$

or, $a_1>2r$

or, $r<\frac{a_1}{2}$

or, $r<\frac{{72}^o}{2}={36}^o$

$\therefore r$ can lie from $0,1,\mathrm{2.....35}$

$\therefore$ No. of tuples $=36$