Question

Let $t_n$ denote the number of integral sided triangle with distinct sides chosen from $\{1,2,3,......n\}$. Then $t_{20}-t_{10}$ equals

• A. $81$
• B. $153$
• C. $163$
• D. $173$

Answer 1

The correct option is A $81$

$t_{20}$ means number of triangle that can be formed using side length $1$ to $20$.

Similarly for $t_{19}$.

$t_{20}-t_{19}$ means number of triangle that can formed with largest length of side $20$.

We have now largest side length as $20$.

To form triangle with this maximum length, minimum length of middle side must be $11$.

Let length of smallest side be $x$, then smallest side will vary from $21-x$ to $x-1$.

Now we have to change $x$ from $11$ to $19$.

By applying these two conditions, we get

$1+3+5+7+9+11+13+15+17=81$