Final answer:
The total number of unique handshakes for 10 people sitting around a circular table, excluding adjacent handshakes, is 35. This is calculated by taking the combinations of 10 people taken 2 at a time (45) and subtracting the number of adjacent handshakes (10).
Step-by-step explanation:
To find the number of handshakes for 10 people sitting around a circular table, where each person shakes hands with everyone else except the adjacent persons, we can use the concept of combinations in mathematics.
Since each person cannot shake hands with the two people adjacent to them, each person will shake hands with 10 - 1 - 2 = 7 people (excluding themselves and the two adjacent persons).
Since a handshake involves two people, every handshake is counted twice when considering each person individually. Therefore, to find the total number of unique handshakes, we can calculate the combinations of 10 people taken 2 at a time and subtract the number of adjacent handshakes, which is the same as the total number of people (10).
The formula for combinations is C(n, k) = n! / (k!(n - k)!), where n is the total number of items and k is the number of items to choose. So, the total number of handshakes is C(10, 2) - 10 = 45 - 10 = 35 handshakes.