Q-2:a)row for a concert. In how many different ways can they be seated:i. Without any restrictions?ii. If each couple is to sit together?iii. If all the men sit together to the right of all the women?b)novels, 3 books of poems, and a dictionary, what is the probability that:i. the dictionary is selected?ii. 2 novels and 1 book of poems are selected?c)studied mathematics, 69 studied history, and 35 studied bothmathematics and history. If one of these students is selected at random,find the probability thati. the student did not take either of these subjects?ii. the student took history but not mathematics?Four married couples have bought 8 seats in the sameIf 3 books are picked at random from a shelf containing 5In a high school graduating class of 100 students, 54 Q-1:a)Justify your answer:i. 3n(n? = n(mod 10)), domain is the set of positive integers.ii. Ja((a < 1) A (a² > 1)), domain is the set of integers.Determine whether each of the following is TRUE or FALSE,iii. Vm+ (m div 3) ), domain is the set of integers.3Using the encryption function f (x) = (10-x) mod 26,0 < x <b)25, to decrypt the message "“DKPG K XCIG HKM".c)Let A, B be any two sets. Prove that if A C B, then A n B = p.

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Answered: Q-2: a) Four married couples have…

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter2: Systems Of Linear Equations

Section2.4: Applications

Problem 29EQ

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Suppose that the check digit is computed as described in Example . Prove that transposition errors of adjacent digits will not be detected unless one of the digits is the check digit. Example Using Check Digits Many companies use check digits for security purposes or for error detection. For example, an the digit may be appended to a -bit identification number to obtain the -digit invoice number of the form where the th bit, , is the check digit, computed as . If congruence modulo is used, then the check digit for an identification number . Thus the complete correct invoice number would appear as . If the invoice number were used instead and checked, an error would be detected, since .
Rework Example 5 by breaking the message into two-digit blocks instead of three-digit blocks. What is the enciphered message using the two-digit blocks? Example 5: RSA Public Key Cryptosystem We first choose two primes (which are to be kept secret): p=17, and q=43. Then we compute m (which is to be made public): m=pq=1743=731. Next we choose e (to be made public), where e must be relatively prime to (p1)(q1)=1642=672. Suppose we take e=205. The Euclidean Algorithm can be used to verify that (205,672)=1. Then d is determined by the equation 1=205dmod672 Using the Euclidean Algorithm, we find d=613 (which is kept secret). The mapping f:AA, where A=0,1,2,...,730, defined by f(x)=x205mod731 is used to encrypt a message. Then the inverse mapping g:AA, defined by g(x)=x613mod731 can be used to recover the original message. Using the 27-letter alphabet as in Examples 2 and 3, the plaintext message no problem is translated into the message as follows: plaintext:noproblemmessage:13142615171401110412 The message becomes 13142615171401110412. This message must be broken into blocks mi, each of which is contained in A. If we choose three-digit blocks, each block mim=731. mi:13142615171401110412f(mi)=mi205mod731=ci:082715376459551593320 The enciphered message becomes 082715376459551593320 where we choose to report each ci with three digits by appending any leading zeros as necessary. To decipher the message, one must know the secret key d=613 and apply the inverse mapping g to each enciphered message block ci=f(mi): ci:082715376459551593320g(ci)=ci613mod731:13142615171401110412 Finally, by re-breaking the message back into two-digit blocks, one can translate it back into plaintext. Three-digitblockmessage:13142615171401110412Two-digitblockmessage:13142615171401110412Plaintext:noproblem The RSA Public Key Cipher is an example of an exponentiation cipher.

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Q-2: a) Four married couples have bought 8 seats in the same row for a concert. In how many different ways can they be seated: i. Without any restrictions? ii. If each couple is to sit together? iii. If all the men sit together to the right of all the women?