20 guests go to the party, so if every other person shakes hands, that’s 10 guests who will end up shaking hands.

Each of these 10 guests will shake hands with 19 other people in the party, so there will be 10 * 19 = 190 handshakes at the party.

Therefore, there will be 190 handshakes at the party.

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Oh sorry, maybe I misinterpreted your question…

If every other person amongst 20 guests shakes hands with just ONE other guest, that would be 10 * 1 = 10 handshakes in total.

In that case, there would be 10 handshakes.

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A formula would only be useful if the number of guests was unspecified. For example, let the number of guests going to the party be: x.

In this case, if every other guest will shake hands, then there will be (x/2) guests who will be shaking hands.

If every other guest will be shaking hands with all the other guests, then the expression would look like this:

(x/2)(x-1) = (1/2)(x^2 – x) or (x^2 / 2) – (x / 2)

If every other guest will be shaking hands with just one other guest, the expression would look like this:

(x/2) * 1 = x / 2.

Hope this helps!