Solutions - 5070-08

(a)

Apply the permutation formula (no repeats allowed):

$$\begin{array}{c}{(26)}_3\cdot {(10)}_4=26\cdot 25\cdot 24\cdot 10\cdot 9\cdot 8\cdot 7\\ \\ \gg \gg \mathrm{78,624,000}\end{array}$$

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(b)

(1) Count letter arrangements (repeats allowed):

${26}^3=\mathrm{17,576}$

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(2) Count digit arrangements with exactly two 9s:

Choose positions for the two 9s: $\left(\genfrac{}{}{0px}{}{4}{2}\right)=6$. Remaining 2 digits from $\{0,\dots ,8\}$: $9^2=81$.

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(3) Combine by multiplication principle:

$$\begin{array}{c}\mathrm{17,576}\cdot 6\cdot 81\gg \gg \mathrm{8,541,936}\end{array}$$