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Number Theory

# Integer Equations - Stars and Bars

A vote to elect the class president is being held in a classroom with $$8$$ students. There are $$3$$ nominees, who do not get to vote. If the vote is a secret vote, i.e. the people do not know who voted for whom, what is the number of possible vote count results?

How many ordered sets of non-negative integers $$(a, b, c, d)$$ are there such that $$a + b +c+d =13$$?

Consider the two sets $$A=\{1,2,3,4\}$$ and $$B=\{5,6,7,8\}.$$ How many possible functions $$f:A\rightarrow B$$ are there such that if $$i<j$$ then $$f(i)\le f(j)?$$

Calvin took a trip to the store to purchase some lightbulbs. His choices at the store were $$25W, 40W, 60W, \mbox{ and } 100W$$ lightbulbs. If Calvin purchased $$13$$ lightbulbs, how many different combinations of bulbs could he have purchased?

A vote to elect the class president is being held in a classroom with $$10$$ students. There are $$3$$ nominees, who do not get to vote. If the vote is a secret vote, i.e. the people do not know who voted for whom, what is the number of possible vote count results?

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