1. Alice is walking on { starting from the origin O . In every second she walks I stop to the left or to the right . How many different walks can Alice take provided she returns to the origin after one minute and stays there?" 5. Alice is now walking on the grid {` starting from the point ( 0. 0 ) . She can walk either up or right . [a ] How many different walk can Alice take to get to the point ( 1 , m ) ?" (b ) USE Your answer to give a combinatorial proof for the following equality : 27 + 2\\ 2 7 " 212 \\ 7 + 1 = 2 . + In + 1` +2. In this question we want to prove that for every integers ! _ _ _ M. [ 1 - 1^ (` ) U = ( 1 ) & = 1 1 in two ways . ( a ) Use Newton's binomial formula ( I + 1j' _ _ _ _ ( ^ ^ ^ ^ ^ to prove ( ! ) ( 6 ) Prove that for every subject A { | 1 . .... " ), | A| is even if | AN [ 1 ] is odd. ( c ) Use 2. I bj to prove that the number of subsets of | 1 ..... it of even size is equal to the number of subjects of (1 . .... it of odd size . ( d) Prove that equation ( I ) holds using the statement in 2. 1 0 )