seven people meet and shake hands with eachother- how many handshakes occur?
(second question)
using inductive reasoning, write a formula for the number of handshakes if the number of people is n
combinations of n objects taken r at a time
C(n,r) = n!/[r!(n-r!)]
C(7,2) = 7!/[2!(5!)]
=7*3
=21
ok, reasoning
first person shakes with 6
second person with 5
third with 4
.
.
.
6th with 1
so
6+5+4+3+2+1 = 21 sure enough
To determine the number of handshakes when seven people meet and shake hands with each other, we can use a combination or factorial approach.
To visualize this, let's assign each person a number from 1 to 7. Person 1 can shake hands with six other people (2 through 7). Person 2 can shake hands with five remaining people (3 through 7), and so on. Hence, the total number of handshakes can be calculated by adding the number of possible handshakes for each person:
Total handshakes = 6 + 5 + 4 + 3 + 2 + 1 = 21.
Now, let's derive a formula for the number of handshakes using inductive reasoning.
Suppose we have n people. Person 1 can shake hands with (n-1) other people (Person 2 through Person n). Person 2 can shake hands with (n-2) remaining people, Person 3 can shake hands with (n-3) remaining people, and so on until Person (n-1) shakes hands with 1 remaining person.
Following this pattern, the total number of handshakes can be calculated by adding the numbers from 1 to (n-1). We can use the formula for the sum of an arithmetic series to simplify this calculation.
Sum of an arithmetic series = (first term + last term) * number of terms / 2
In this case, the first term is 1, the last term is (n-1), and the number of terms is (n-1).
Total handshakes = ((1 + (n-1)) * (n-1)) / 2
Simplifying this formula further, we have:
Total handshakes = (n * (n-1)) / 2
So, if you want to know the number of handshakes when the number of people is n, you can use this formula: Total handshakes = (n * (n-1)) / 2.