Independence Day Celebration

On analyzing the total no of handshakes, let there be $x+1$ no of people. Each people handshake with each other only once, So the people who already had handshake with, won't do it again. Therefore, we can conclude that

$\dfrac{x(x-1)}{2} = 210 \\ \implies x(x-1) = 420\\ \implies x^2 - x - 420 = 0 \implies x^2 - 21x + 20x - 420 = 0 \\ \implies (x-21)(x+20) = 0 \\ \implies x = 21, x = -20 \\ \implies x = 21$

Therefore there are, $x = 21$ , 21 People in the room