# Ministry of Education are inviting tender for four categories promoting the use of IT in education.

Ministry of Education are inviting tender for four categories promoting the use of IT in education. Each category consists of 5, 4, 3, and 7 projects, respectively. Each project appears on exactly one category. How many possible projects are there to choose from? Explain your answer.
My Answer: $\left(5+4\right)+\left(4+4\right)+\left(3+3\right)+\left(7+4\right)=9+8+6+11=34$ possible projects to choose from. I used the sum rule here.
Is this correct?
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Frederick Greer
Since each project appears in exactly one category, the four lists of projects can be combined into one list of projects containing $5+4+3+7=19$ projects. Hence, there are 19 projects from which to choose.
The Addition Principle (or Sum Rule) states that if there are ${n}_{1}$ ways of performing one task and ${n}_{2}$ ways of performing another task that cannot be performed at the same time, there are ${n}_{1}+{n}_{2}$ ways of performing both tasks.
In this case, we can select a project from category one in five ways, a project from category two in four ways, a project from category three in three ways, or a project in category four in seven ways. Since it is not possible to choose the same project from more than one list, the Addition Principle applies. Hence, there is a total of $5+4+3+7=19$ projects from which to choose.