Discrete Mathematics
Q21.
Let A be a sequence of 8 distinct integers sorted in ascending order. How many distinct pairs of sequences, B and C are there such that (i) each is sorted in ascending order, (ii) B has 5 and C has 3 elements, and (iii) the result of merging B and C gives A?Q22.
The minimum number of cards to be dealt from an arbitrarily shuffled deck of 52 cards to guarantee that three cards are from same suit isQ23.
In a permutation a_1.....a_n of n distinct integers, an inversion is a pair (a_i, a_j) such that i \lt j and a_i \gt a_j. If all permutation are equally likely, what is the expected number of inversions in a randomly chosen permutation of 1.....n?Q24.
Mala has a colouring book in which each English letter is drawn two times. She wants to paint each of these 52 prints with one of k colours, such that he colour pairs used to colour any two letters are different. Both prints of a letter can also be coloured with the same colour. What is the minimum value of k that satisfies this requirement?Q25.
How many sub strings of different lengths (non-zero) can be formed from a character string of length n?Q26.
Let \Sigma = (a, b, c, d, e) be an alphabet. We define an encoding scheme as follows : g(a) = 3, g(b) = 5, g(c) = 7, g(d) = 9, g(e) = 11. Let p_{i} denote the i-th prime number (p_{i}=2). For a non-empty string s=a_{1}...a_{n}, where each a_{i} \in \Sigma, define f(i)=\prod_{i=1}^{n}{P_{i}}^{g(a_{i})}. For a non-empty sequence < s_{j},...,s_{n} > of strings from \Sigma^{+}, define h(<s_{i}...s_{n} >)=\prod_{i=1}^{n}{P_{i}}^{f(s_{i})} Which of the following numbers is the encoding, h, of a non-empty sequence of strings?Q27.
In how many ways can we distribute 5 distinct balls, B_1, B_2, \ldots, B_5 in 5 distinct cells, C_1, C_2, \ldots, C_5 such that Ball B_i is not in cell C_i, \forall i= 1,2,\ldots 5 and each cell contains exactly one ball?Q28.
Let f:A\rightarrow B be an onto (or surjective) function, where A and B are nonempty sets. Define an equivalence relation \sim on the set A as a_1\sim a_2 \text{ if } f(a_1)=f(a_2), where a_1, a_2 \in A . Let \varepsilon =\{[x]:x \in A\} be the set of all the equivalence classes under \sim . Define a new mapping F: \varepsilon \rightarrow B as F([x]) = f(x), for all the equivalence classes [x] in \varepsilonWhich of the following statements is/are TRUE?Q29.
Which one of the following is the closed form for the generating function of the sequence \{a_n \}_{n \geq 0} defined below?a_n= \left \{ \begin{matrix} n+1, &n \text{ is odd} \\ 1,& \text{otherwise} \end{matrix} \right.Q30.
Consider the following sets, where n\geq 2: S1: Set of all nxn matrices with entries from the set \{a, b, c\} S2: Set of all functions from the set\{0,1,2,...,n^2-1\} to the set \{0, 1, 2 \} Which of the following choice(s) is/are correct?[MSQ]