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Linear Diophantine Equations

What amounts of money can you make with only $5 and $7 bills? Can you make $23 with those 2 types of bills?

With Restriction

         

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 \(a_1< a_2<a_3\le a_4\le a_5?\)

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 \(a_1\le a_2<a_3\le a_4\le a_5\le a_6?\)

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