SOLUTION: In a student union meeting in a school, 16 students show up.Each shakes hands with each other exactly once.Determine the total number of handshakes.
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Question 1087253: In a student union meeting in a school, 16 students show up.Each shakes hands with each other exactly once.Determine the total number of handshakes.
You can put this solution on YOUR website! There are 16 students. Any given student has 15 people to shake hands with. Doing this 16 times means we have 16*15 = 240 orderings.
Keep in mind that instructions state "Each shakes hands with each other exactly once".
So we need to get rid of the duplicate handshakes. For example, let's say we have student A and student B. One handshake is AB while another handshake is BA. The order doesn't matter as it's the same group.
To correct this, we divide by 2 and it gets us 240/2 = 120 which is the final answer.
Here is a table showing all the possible handshakes. Along the left and top borders are the student names (code name letters A through P, which is the 16th letter of the alphabet).
Combos like BB or CC are impossible so we ignore anything along the main diagonal. Those are marked with X. Anything below the diagonal is a mirror of what is above the diagonal. This is where the double-counting comes in. So we simply pick one side. I'm going to pick the stuff above the diagonal. Anything below the diagonal will have X in it. The boxes with S in it indicate a handshake. If you carefully count out the S letters, you'll find that there are 120 of them. An alternative is to sum the row counts to get: 15+14+13+...+3+2+1 and you'll get 15*(15+1)/2 = 15*16/2 = 120 which is the same as before.
Using the formula is the preferred method since the table is a bit tedious.
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