Consider the following statement: "We have a group of people consisting of 6 men and 5 women. Some people greet each other with a handshake. Prove that if 28 handshakes were exchanged in total, then at least one woman shaked hands with some man".

The proof below contains some missing phrases. From the lists below, choose correct phrases to form a complete and correct proof.

Proof: We will estimate how many man-to-man handshakes and woman-to-woman handshakes can be exchanged. Since there are 6 men, the number of man-to-man handshakes will be (Phrase 1). Since there are 5 women, the number of woman-to-woman handshakes will be (Phrase 2). Therefore the total number of same-gender handshakes is (Phrase 3). Since the total number of handshakes is 28, and (Phrase 4), at least one woman must have shaked hands with a man. QED

Choose a correct Phrase 1:

at most = 15

at most 6² = 36

at most 65 = 30

at least = 15

Choose a correct Phrase 2:

equal = 10

at most 5² = 25

at most = 10

at least = 10

Choose a correct Phrase 3:

at most = 55

at most 65 = 30

at least 25

at most 25

Choose a correct Phrase 4:

15

28

6 > 5

28 > 25

6 2 6 2 5 2 11 2