The Handshaking question

you are choosing 2 people from "n" people

it's like asking
if you have 7 books and you only have room for 2 in your bag. How many ways can you choose 2 from 7 books

we will call the books
ABCDEFG

you can bring
AB
AC
AD
AE
AF
AG
(BA would be the same as bringing AB, you don't count it again)
BC
BD
BE
BF
BG
CD
CE
CF
CG
DE
DF
DG
EF
EG
FG
that's 21 different ways to choose 2 from 7

it's the same when shaking hands
once one person has shaken hands with another, they don't shake again

nCr(7 , 2)
or
7C2

is
7! / [2!(7 - 2)!]

! is the symbol for factorial
7! = 7 * 6 * 5 * 4 * 3 * 2 * 1

(7 * 6 * 5 * 4 * 3 * 2 * 1) / (2 * 1 * 5 * 4 * 3 * 2 * 1)
(7 * 6) / 2
42 / 2
21

if there were 100 people
100! / [2!(100 - 2)!]
100! / (2! * 98!)
(100 * 99) / 2
4950 handshakes

n people

n! / [2!(n - 2)!]

1. 1 + 2 + 3 + . . . + 99 = 99 * 100 / 2 = 4950

2. 1 + 2 + 3 + . . . + (n-1) = (n-1) n / 2

Well known formula:

S = 1 + 2 + . . . + (k-1) + k <----- k terms
S = k + (k-1) + . . . + 2 + 1 <----- k terms
----------------------------------
2S = (k+1) + (k+1) + . . . + (k+1) + (k+1) <----- k terms
2S = k (k+1)
S = k (k+1) / 2

So when we add from 1 to (n-1), we get
S = (n-1) ((n-1)+1) / 2 = (n-1) n / 2

=========================

Alternatively

1. (100 C 2) pronounced "100 choose 2" = 100! / (2! 98!) = 4950

2. (n C 2) = n! / (2! (n-2!)) = n * (n-1) * (n-2)! / (2! (n-2)!) = n (n-1) / 2! = n (n-1) / 2

-- Ματπmφm --