+92 How many 10-digit binary numbers start with 2 ones or end with 2 ones (or both)?
+505 The number of 10-digit binary numbers that end with 2 ones is \(2^7=128\), since the first digit must be 1, the last 2 digits must be 1, and the middle 7 digits can be either 1 or 0.
The number of 10 digit binary numbers that start with 2 ones is \(2^8 = 256\), since the first 2 digits must be 1 and the other 8 digits can be either 1 or 0.
If you add the 2 numbers together, you get \(128+256 = 384\), but that is not the answer because the number of 10-digit binary numbers satisfying both conditions is overcounted and needs to be subtracted.
The number of 10-digit binary numbers that satisfy both conditions is \(2^6 = 64\), since the first 2 and the last 2 digits must be equal to 1 and the middle 6 digits can be equal to either 1 or 0.
Therefore, the answer is \(384-64=\boxed{320}\)
