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Discrete Math: Inclusion-Exclusion & Recurrence Relations, Assignments of Mathematics for Computing

Biju Patnaik University of TechnologyMathematics for Computing

Principle of inclusion and exclusion Recurrence relations

Typology: Assignments

2020/2021

Uploaded on 02/21/2021

ajay-singh-23
ajay-singh-23 🇮🇳

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Discrete Mathematics
Problem Sheet 3
Principle of Inclusion - Exclusion:
1. A certain computer center employs 100 computer programmers. Of these
47 can program in FORTRAN, 35 in Pascal and 23 can program in both
languages. How many can program in neither of these 2 languages?
2. Suppose that, in addition to the information given in 1, there are 20
employees that can program in COBOL, 12 in COBOL, 12 in COBOL
and FORTRAN, 11 in Pascal and COBOL and 5 in FORTRAN, Pascal,
and COBOL. How many can program in none of these 3 languages.
3. Find the number of integers between 1 and 1,000 inclusive that are divisible
by none of 5,6, and 8. Note that the intersection of the set of integers
divisible by 6 with the set of integers divisible by 8 is the set of integers
divisible by 24.
4. Eight people enter an elevator at the first floor. The elevator discharges
passengers on each successive floor until it empties on the fifth floor. How
many different ways can this happen?
5. How many arrangements are there of MISSISSIPPI with no pair of con-
secutive letters the same?
6. Suppose that a person with 10 friends invites a different subset of 3 friends
to dinner every night for 10 days. How many ways can this be done so
that all friends are included at least once?
7. How many arrangements are there of 3.a’s, 3.b’s and 3.c’s
(a) without 3 consecutive letters the same?
(b) having no adjacent letters the same?
8. Use the principle of inclusion exclusion to count the number of primes
between 41 and 100 inclusive.
9. How many 6-digit decimal numbers contain exactly three different digits?
Recurrence Relations:
1. Find the coefficient of X16 in (1 + X4+X8)10.
2. Find the generating function for the number of r-combinations of {3.a,
5.b, 2.c}.
3. Build a generating function for determining the number of ways of making
change for a dollar bill in pennies, nickels, dimes, quarters, and half-dollar
pieces. Which coefficient do we want?
1
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Discrete Mathematics

Problem Sheet 3

Principle of Inclusion - Exclusion:

  1. A certain computer center employs 100 computer programmers. Of these 47 can program in FORTRAN, 35 in Pascal and 23 can program in both languages. How many can program in neither of these 2 languages?
  2. Suppose that, in addition to the information given in 1, there are 20 employees that can program in COBOL, 12 in COBOL, 12 in COBOL and FORTRAN, 11 in Pascal and COBOL and 5 in FORTRAN, Pascal, and COBOL. How many can program in none of these 3 languages.
  3. Find the number of integers between 1 and 1,000 inclusive that are divisible by none of 5,6, and 8. Note that the intersection of the set of integers divisible by 6 with the set of integers divisible by 8 is the set of integers divisible by 24.
  4. Eight people enter an elevator at the first floor. The elevator discharges passengers on each successive floor until it empties on the fifth floor. How many different ways can this happen?
  5. How many arrangements are there of MISSISSIPPI with no pair of con- secutive letters the same?
  6. Suppose that a person with 10 friends invites a different subset of 3 friends to dinner every night for 10 days. How many ways can this be done so that all friends are included at least once?
  7. How many arrangements are there of 3.a’s, 3.b’s and 3.c’s (a) without 3 consecutive letters the same? (b) having no adjacent letters the same?
  8. Use the principle of inclusion exclusion to count the number of primes between 41 and 100 inclusive.
  9. How many 6-digit decimal numbers contain exactly three different digits?

Recurrence Relations:

  1. Find the coefficient of X^16 in (1 + X^4 + X^8 )^10.
  2. Find the generating function for the number of r-combinations of {3.a, 5.b, 2.c}.
  3. Build a generating function for determining the number of ways of making change for a dollar bill in pennies, nickels, dimes, quarters, and half-dollar pieces. Which coefficient do we want?
  1. Use partial fractions to compute:

(a)

1 − 7 X + 12X^2

(b)

7 X^2 + 3X + 2

(X − 2)(X + 1)^2

(c)

1 − 7 X + 3X^2

(1 − 3 X)(1 − 2 X)(1 + X)

  1. Find the coefficient of X^12 in

1 − X^4 − X^7 + X^11 (1 − X)^5

  1. Find the coefficient of X^14 in

(a) (1 + X +... + X^8 )^10 (b) (X^2 + X^3 +... + X^7 )^4

  1. How many ways are there to place an order of 12 chocolate sundaes if there are 5 types of sundaes, and at most 4 sundaes of one type are allowed?
  2. How many ways are there to paint 20 identical rooms in a hotel with 5 colors if there is only enough blue, pink, and green pain to paint 3 rooms?
  3. Write a generating function for an, the number of ways of obtaining the sum n when tossing 9 distinguishable dice. Then find a 25.
  4. In each of the following a recurrence relation and a function are given. In each case, show that the function is a solution of the given recurrence relation. (a) an − an− 1 = 0; an = C. (b) an − an− 1 = 2; an = 2n + C.
  5. Find a recurrence relation for the number of ways to arrange flags on a flagpole n feet tall using 4 types of flags; red flags 2 feet high, or white, blue, and yellow flags each 1 foot high.
  6. Let Pn be the number of permutations of m letters taken n at a time with repetitions but no 3 consecutive letters being the same. Derive a recurrence relation connecting Pn, Pn− 1 , and Pn− 2.
  7. Find the recurrence relation that counts the number of ways of making a selection from the numbers 1, 2 ,... , n such that no integer is more than one place removed from its position in the natural order.
  8. Find a recurrence relation for the number of n-digit binary sequence that have the pattern 010 occurring at the nth digit. For example, the pattern 010 occurs at the fourth and the ninth digits in the sequence