7 people so first one shakes hand with the remaining people. that is 6
The 2nd person also shakes hand with 6 persons but we have already counted his handshake with first person. so we have to add 6-1 = 5 shakes for 2nd person
like wise 3rd person shakes hand with the remaining 6-2 = 4 people
4th person with 6-3=3 people
5th person with 6-4= 2 persons
6th person with 6-5=1 person
and 7th one with no one because already he has shaken his hand with everyone.
since these people shake their hands with others only once we just have to add the obtained values
6+5 +4+3+2+1= 21
totally 21 handshakes will be there in that room