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QuestionHow many handshakes occur when 4 people shake hands once? Generalize for nn people.

Studdy Solution

STEP 1

Assumptions1. Each person shakes hands with every other person exactly once. . A handshake between two people is considered a single handshake, regardless of who initiates it.

STEP 2

We can use the concept of combinations to solve this problem. The number of handshakes is the same as the number of ways to choose2 people out of4 to shake hands.
The formula for combinations isC(n,k)=n!k!(n−k)!C(n, k) = \frac{n!}{k!(n-k)!}where nn is the total number of items, kk is the number of items to choose, and !! denotes factorial.

STEP 3

Now, plug in the given values for nn and kk to calculate the number of handshakes.
C(,2)=!2!(−2)!C(,2) = \frac{!}{2!(-2)!}

STEP 4

implify the factorial expressions.
4!=4times3times2times1=244! =4 \\times3 \\times2 \\times1 =242!=2times1=22! =2 \\times1 =2So, the expression becomesC(4,2)=242times2C(4,2) = \frac{24}{2 \\times2}

STEP 5

Calculate the number of handshakes.
C(4,2)=244=C(4,2) = \frac{24}{4} =So, if four people shake hands with one another exactly once, handshakes take place.
b. Generalize the solution for nn people.

STEP 6

To generalize the solution for nn people, we can use the same formula for combinations. The number of handshakes is the same as the number of ways to choose2 people out of nn to shake hands.
So, the general formula for the number of handshakes isC(n,2)=n!2!(n−2)!C(n,2) = \frac{n!}{2!(n-2)!}

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