With n bits, you can represent 2ⁿ integers, from 0 to (2^n)-1

To write a number in binary, you must use powers of 2.

Examples:

Q: How can I represent 100? A: 64+32+3+1

Q: How can I represent 37? A: 32+4+1

Q: How can I represent 24? A: 16+8

Q: How can I represent 26? A: 16+8+2

Q: How can I represent 200? A: 128+64+8 **

Now how do I convert those powers into binary form?

Easy.

You can see that powers of 2 is the following series:

1+2+4+8+16+32+64+128+256+...

Now to represent your 200 countries, you need 8, 64 and 128.

Q: What exponent do I need to give to 2 to get 8? A: 3

Q: What exponent do I need to give to 2 to get 64? A: 6

Q: What exponent do I need to give to 2 to get 128? A: 7

What we just calculated that the positions of 8, 64 and 128 are 3, 6, and 7 respectively.

Now we need to form a binary numbers, knowing that when forming binary numbers, position ZERO is the rightmost side of the number

Like this:

[p7][p6][p5][p4][p3][p2][p1][p0]

So, we need position 3 6 and 7. Let's do it now. Let's put 1 in each of these positions, and fill the rest with zeroes.

[p7][p6][p5][p4][p3][p2][p1][p0]

[1][1][0][0][1][0][0][0]

Then we verify:

0x2^{0+0x21+0x22+1x23+0x24+0x25+1x26+1x27=200}

Yay! Thus, to answer your answer, recall that with n bits you can represent integers from 0 to 2^n-1. 2ⁿ is a power of 2 so you need the smallest power of 2 greater or equal to 200. The smallest is 256, which is 2^8, which can represent 0 to 255 ((2^8)-1=255