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Chapter 8: Q35SE (page 567)

Students in the school of mathematics at a university major in one or more of the following four areas: applied mathematics (AM), pure mathematics (PM), operations research (OR), and computer science (CS). How many students are in this school if (including joint majors) there are 23 students majoring in AM; 17 in PM; 44 in OR; 63 in CS; 5 in AM and PM; 8 in AM and CS; 4 in AM and OR; 6 in PM and CS; 5 in PM and OR; 14 in OR and CS; 2 in PM, OR, and CS; 2 in AM, OR, and CS; 1 in PM, AM, and OR; 1 in PM, AM, and CS; and 1 in all four fields.

Short Answer

Expert verified

Total number of students in the school is 110.

Step by step solution

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01

Given Information

If there are 23 students majoring Applied Mathematics (AM), 17 in Pure Mathematics (PM), 44 in Operational Research (OR), 63 in Computer Science (CS), 5 in AM and PM, 8 in AM and CS, 4 in AM and OR, 6 in PM and CS, 5 in PM and OR, 14 in OR and CS, 2 in PM, OR and CS, 2 in AM, OR and CS, 1 in PM, AM and OR, 1 in PM, AM, CS and 1 in all four fields.

02

Define Principle of inclusion-exclusion

Principle of Inclusion and Exclusion is an approach which derives the method of finding the number of elements in the union of two finite sets.

03

Calculate the number of students

Let n(AM), n(PM), n(CS), n(OR)be the number of students who are majoring in Applied Mathematics (AM), Pure Mathematics (PM), Computer Sciences (CS), Operational Research (OR) respectively.

So,

By the principle of inclusion-exclusion

Hence, total number of students in the school is

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Most popular questions from this chapter

Use Exercise 31 to show that if a<bd, thenf(n)isO(nd).

Use generating functions to solve the recurrence relation ak=ak-1+2ak-2+2kwith initial conditions a0=4anda1=12.

(a) Set up a divide-and-conquer recurrence relation for the number of multiplications required to computexn, where xis a real number and nis a positive integer, using the recursive algorithm from Exercise 26 in Section 5.4.

b) Use the recurrence relation you found in part (a) to construct a big- Oestimate for the number of multiplications used to compute xnusing the recursive algorithm.

Find the coefficient of \({x^{10}}\) in the power series of each of these functions.

a) \({\left( {1 + {x^5} + {x^{10}} + {x^{15}} + \cdots } \right)^3}\)

b) \({\left( {{x^3} + {x^4} + {x^5} + {x^6} + {x^7} + \cdots } \right)^3}\)

c) \(\left( {{x^4} + {x^5} + {x^6}} \right)\left( {{x^3} + {x^4} + {x^5} + {x^6} + {x^7}} \right)(1 + x + \left. {{x^2} + {x^3} + {x^4} + \cdots } \right)\)

d) \(\left( {{x^2} + {x^4} + {x^6} + {x^8} + \cdots } \right)\left( {{x^3} + {x^6} + {x^9} + } \right. \cdots \left( {{x^4} + {x^8} + {x^{12}} + \cdots } \right)\)

e) \(\left( {1 + {x^2} + {x^4} + {x^6} + {x^8} + \cdots } \right)\left( {1 + {x^4} + {x^8} + {x^{12}} + } \right. \cdots )\left( {1 + {x^6} + {x^{12}} + {x^{18}} + \cdots } \right)\)

Find the coefficient of \({x^{12}}\) in the power series of each of these functions.

a) \(1/(1 + 3x)\)

b) \(1/(1 + 3x)\)

c) \(1/{(1 + x)^8}\)

d) \(1/{(1 - 4x)^3}\)

e) \({x^3}/{(1 + 4x)^2}\)
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