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Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2
(a) Evaluate $(x + x^2 + x^3 + x^4)^2$.

(b) Imagine you have two spinners labelled $1, 2, 3$ and $4$ and spin them together. The score is the sum of the results from the two spinners. Find the theoretical frequency distribution of the scores.

(c) What do you notice about this frequency distribution and the coefficients in the polynomial expansion from (a)?

(d) You may like to try the computer simulation. The table gives relative frequencies.

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(e) Notice that the powers in $(x^1 + x^2 + x^3 + x^4)$ correspond to the labels on the spinners. Can you factorize the expression $(x + x^2 + x^3 + x^4)^2$ into two different polynomials which correspond to a re-labelling of the spinners, so each has four non-negative integer labels, giving new pairs of spinners with the same frequency distribution of scores? This re-labelling can be done in more than one way.

(f) You may like to run the computer experiment with the labellings you have found to see if they do produce the expected relative frequencies of the scores. (You can change the numbers on the spinners by clicking on the numbers.)

(g) Using the method from part (e), find other pairs of spinners which can be re-labelled in more than one way to give the same frequency distribution of scores.

What about a $2$-spinner and a $3$-spinner? What about two ordinary dice?