Number Theory

Linear Diophantine Equations

Integer Equations - With Restriction

         

Ben throws a fair die five times. If we label the outcome of the nthn^{\text{th}} throw as an,a_n, what is the number of cases such that a1<a2<a3a4a5?a_1< a_2<a_3\le a_4\le a_5?

Consider the two sets A={1,2,3,4,5,6}A=\{1,2,3,4,5,6\} and B={7,8,9,10}.B=\{7,8,9,10\}. How many functions f:ABf:A\rightarrow B are there that satisfy f(1)f(2)f(3)<f(4)f(5)f(6)?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?9?

How many 4-digit numbers are there such that the digit sum is 11?11?

Ben throws a fair die six times. If we label the outcome of the nthn^{\text{th}} throw as an,a_n, what is the number of cases such that a1a2<a3a4a5a6?a_1\le a_2<a_3\le a_4\le a_5\le a_6?

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