Number Theory
Linear Diophantine Equations
Ben throws a fair die five times. If we label the outcome of the \(n^{\text{th}}\) throw as \(a_n,\) what is the number of cases such that \(a_1< a_2<a_3\le a_4\le a_5?\)
Consider the two sets \(A=\{1,2,3,4,5,6\}\) and \(B=\{7,8,9,10\}.\) How many functions \(f:A\rightarrow B\) are there that satisfy \(f(1)\le f(2)\le f(3)<f(4)\le f(5)\le f(6)?\)
How many 4-digit numbers are there such that the digit sum is \(9?\)
How many 4-digit numbers are there such that the digit sum is \(11?\)
Ben throws a fair die six times. If we label the outcome of the \(n^{\text{th}}\) throw as \(a_n,\) what is the number of cases such that \(a_1\le a_2<a_3\le a_4\le a_5\le a_6?\)