Linear Diophantine Equations
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?