STEP I 1990 question 2 solution

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Note that if you add the three together, the coefficients of all the binomial coefficients, , except for those where k is a multiple of three, are equal to . When k is a multiple of three, the coefficient is three.

So

As by the sum to product formulae.

.

.

Due to the symmetry of the binomial coefficients: , and so on.

Solution by Dystopia.