Number Theory

Linear Diophantine Equations

Integer Equations - Stars and Bars

         

A vote to elect the class president is being held in a classroom with 88 students. There are 33 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) (a, b, c, d) are there such that a+b+c+d=13 a + b +c+d =13?

Consider the two sets A={1,2,3,4}A=\{1,2,3,4\} and B={5,6,7,8}.B=\{5,6,7,8\}. How many possible functions f:ABf:A\rightarrow B are there such that if i<ji<j then f(i)f(j)?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, and 100W25W, 40W, 60W, \mbox{ and } 100W lightbulbs. If Calvin purchased 1313 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 1010 students. There are 33 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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