15 boys and 15 girls attended a party. Each boy shook hands with exactly five other people, and each girl shook hands with exactly seven other people. How many handshakes took place?
视频信息
答案文本
视频字幕
We have a handshake problem at a party. There are 15 boys and 15 girls. Each boy shakes hands with exactly 5 other people, and each girl shakes hands with exactly 7 other people. We need to find the total number of handshakes that took place at the party.
To solve this problem, we need to understand a key principle about counting handshakes. Each handshake involves exactly two people. When we count handshakes from each person's perspective, we double-count every handshake because each handshake gets counted once from person A's side and once from person B's side. Therefore, the total number of unique handshakes equals the sum of all individual counts divided by two.
Now let's calculate the individual contributions. First, we count handshakes from the boys' perspective: 15 boys times 5 handshakes each equals 75. Next, we count from the girls' perspective: 15 girls times 7 handshakes each equals 105. Adding these together: 75 plus 105 equals 180. This gives us the sum of all individual handshake counts, but remember this includes double-counting.
Now we apply our formula to find the final answer. Total handshakes equals 180 divided by 2, which equals 90. Let's verify this makes sense: with 30 people total, the maximum possible handshakes would be 30 times 29 divided by 2, which equals 435. Our answer of 90 is well within reasonable bounds. Therefore, 90 handshakes took place at the party.