chapter  7
2 Gregory Galperin and Hillel Gauchman’s Problem
Pages 2

In 2004 Gregory Galperin and Hillel Gauchman posed the problem of proving the following identity [Galperin and Gauchman 04]:

1 k ( n k

) = 1 2n−1

) k

. (7.6)

By (7.4) the left side is

1 k ( n k

) = 1 n

1( n−1 k−1 ) = 2−n n∑

2k

k =

1 2n−1

2k−1

k . (7.7)

Now consider the sum on the right side of (7.6). The difference in n is

) k

− ∑

) k

= ∑

( n−1 k−1 )

k =

1 n

( n

k

) .