Express 3x4+2x3 - 2x2+x in terms of [x]4, [x]3, [x]2 and [x].
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Which of the following statements are true and which are false? Give reasons for your answer.
i) ( ~p v q) and (p q) are logically equivalent.
ii) (m+n)! > m!+n! for m, n any positive integers.
iii) The number of distributions of m indistinguishable objects into n distinguishable containers is the same as the number of non-negative integral solutions of the equation x1+x2+....+xn = m.
iv) Sn, the number of subsets of a set with n elements, satisfies a second order linear homogeneous recurrence equation with constant coefficients.
v) K3,4 is Hamiltonian.
Write (x z)
(y
z') in CNF and DNF.
Give examples of trivially true and vacuously true statements.
Solve the recurrence
6an+3-13an+2+9an+1-2an = 2-n;n > 0; if a0 = 1;a1 = 1;a2 = 1
If an apple costsRs.15, a papaya costs Rs.17 and a banana costs Rs.8, write the generating function for the number of ways of purchasing these fruits with Rs.n.
A die is rolled twice and the sum of the numbers that appear is observed. What is the probability that the sum is either a perfect square or a perfect cube?
If a planar graph has the degree sequence {2;2;3;3;4;4;4}, how many faces will it have? Draw a planar graph with this degree sequence and number the faces to check your answer.