You are given a balance and one copy of each of ten weights of 1,2,4,8,16,32,64,128,256 and 512 grams. An object weighing M grams, where M is a positive integer, is put on one of the pans and may be balanced in different ways by placing various combinations of the given weights on either pan of the balance.
(a) Prove that no object may be balanced in more than 89 ways.
(b) Find a value of M such that an object weighing M grams can be balanced in 89 ways.(A Shapovalov, A Kulakov) combinatoricsweighingsnumber theory