DURHAM UNIVERSITY — Department of Mathematical Sciences
COLLECTION 2016
Name:
College:
DISCRETE MATHEMATICS
MATH1031
Time allowed: 45 minutes. Answer all questions in the space provided. Approved electronic calculators may
be used. Note: the questions have different marks associated with them. Full marks may be obtained without
the use of a calculator. Answers containing factorials and/or binomial coefficients are fine.
1. A hand of 6 cards is dealt from a standard deck of 52 playing cards. How many hands are possible in which:
(i) The 6 cards are all of the same suit?
(ii) The denominations of the 6 cards form a sequence (e.g. A23456)? Note: Ace can be high or low.
(iii) The 6 cards form three distinct pairs of denominations (e.g. 3388QQ)?
2.
(i) How many arrangements of the letters HEMISPHERE are there?
(ii) In how many of the arrangements in (i) are the two Hs adjacent?
(iii) In how many of the arrangements in (i) are exactly two (but not three) Es adjacent?
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3. How many solutions (x1 , x2 , x3 , x4 ) are there to the equation
x1 + x2 + x3 + x4 = 14
where each xi is an integer and xi ≥ 2 − i?
n
X
n 2r
4. Evaluate
r
3 .
r
r=0
Hint: consider (1 + x2 )n .
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5. How many arrangements of the letters XXYYZZ have no pair of adjacent letters the same?
6. Solve the recurrence relation an = 2an−1 + n, where a0 = 1.
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7. Let pn denote the number of ways in which n identical beads may be distributed among n identical boxes.
(i) Compute p3 .
(ii) Suppose three beads, all of different colours, are distributed among three identical boxes. How many
ways of distributing the beads are there?
(iii) Write down a generating function for (pn ).
Hint: Use your work for (i) to help with (ii).