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(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
(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