# If a die is rolled 35 times, there are 6^{35}

If a die is rolled $$\displaystyle{35}$$ times, there are $$\displaystyle{6}^{{{35}}}$$ different sequences possible.
What fraction of these sequences have exactly five 1s?

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Deufemiak7
Die contains six possible outcomes $$\displaystyle{\left[{1},{2},{3},{4},{5},{6}\right]}$$. The number of possible outcomes when a die is rolled 35 times is $$\displaystyle{6}^{{{35}}}$$
There would be 35 slots when it is rolled 35 times. The sequence must consist of five 1's
This can be done in $$\displaystyle{35}{C}_{{5}}$$ ways. After filling of five slots with 1's there would be 30 remaining slots. They can be filled with numbers $$\displaystyle{2},{3},{4},{5},{6}$$ in $$\displaystyle{5}^{{{30}}}$$ ways
The fraction of $$\displaystyle{6}^{{{35}}}$$ sequences that have exactlyfive 1's is:
$$\displaystyle{P}{r}{o}{b}{a}{b}{i}{l}{i}{t}{y}={\frac{{{35}{C}_{{5}}\times{5}^{{{30}}}}}{{{6}^{{{35}}}}}}$$
$$\displaystyle={0.1759}$$