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The correct OPTION is (B) hyperplane

Easy explanation: In an m-dimensional space, every linear equation produces a hyperplane for n variables. The SOLUTION set is the intersection of these hyperplanes and is planar which may have any DIMENSION smaller than m.

The CORRECT choice is (C) nonhomogeneous

Easy explanation: A LINEAR system Cx = d is known as a homogeneous system if d! = 0. The homogeneous linear system Ax = 0 is called its CORRESPONDING homogeneous linear system.

Correct choice is (d) Bk

The explanation: To solve the PROBLEM FIRST find the prime FACTORIZATION of each term of the product, and place the factors of each term into a box. Then, since N is the product of distinct prime factors, each prime factor appears in a unique box. Since the product of all of these terms is N, each prime factor must be in a box. Conversely, for any arrangement of these n distinct primes into r IDENTICAL boxes, multiply the primes in a box to create a term and the product of these terms results in N. This establishes the BIJECTION and the number of ways is Bk which is Bell number.

The correct OPTION is (b) 6435

For explanation: This problem can be viewed as identical objects distributed into DISTINCT non-empty BINS. Using the formula for these kind of DISTRIBUTIONS ^n-1Cr-1 = ^15C7 = 6435. THUS, there are distributions of the pizza slices.

Right answer is (d) ^13C9

To EXPLAIN: For this type of problem, n>=r MUST be true and so according to stars and bars model, the number of possible arrangements of stars and bars is n-1Cr-1 or equivalently, there are ^n-1Cr-1distributions of n identical objects into r DISTINCT non-empty BINS. In this example, there are n = 14 identical objects to be distributed among r=10 distinct bins. Using the above FORMULA, the number of possible distributions is ^13C9.

The correct answer is (d) 855

For explanation: The problem can be DESCRIBED as distinct objects into any number of identical bins and this number can be found with B7 = ∑S(7,k), where S(7,k) is the number of distributions of 5 distinct objects into k identical non-empty bins, so that S(7,1) = 1, S(7,2) = 63, S(7,3) = 301, S(7,4) = 350 and S(7,5) = 140. These values can be found using the recurrence relation identity for Stirling numbers of the second KIND. Thus, B7 = 1 + 63 + 301 + 350 + 140 = 855.

Right ANSWER is (B) ^20P6

To elaborate: This is a permutation problem since the chocolates are distinct. The answer is P(20, 6) -> the number of ways to arrange 20 things taken 6 at a TIME -> which is \(\frac{20!}{(20-6)!}\) = 20*19*18*17*16*15.

Right ANSWER is (d) 2 pm

The BEST explanation: DIVIDE 146 with 24. The REMAINDER is the time on the 24 hour clock. So, 146 = 6*24 + 2 and the result is 2pm.

Correct answer is (b) ^64C20

To explain I would say: IMAGINE the 20 cookies ONE is choosing are indistinguishable dots. The 45 different types of cookies are like 45 DISTINGUISHABLE boxes and so the answer is C(45 + 20-1, 20) = ^64C20.

The correct choice is (c) 966

For explanation I would say: Using the inclusion-exclusion principle, the total number of WAYS of SPLITTING the GIRLS into 3 groups is 3^8 + 3.(2^8) + 3.(1^8). However, SINCE the three groups are identical we need to DIVIDE by 3!. Hence, the answer is 966.

The correct CHOICE is (a) 16

Explanation: To find the NUMBER of even factors, we can multiply the number of even factors by the power of 2. For 5065, we have (3 + 1)(1 + 1)(2) = 4 * 2 * 2 = 16 even factors.

The CORRECT choice is (c) 18102

Explanation: The prime factorization of 1800 is 431 * 2^2 * 5 and

S(2^2) = 1 + 2 + 4 = 7

S(5^2) = 1 + 5 = 6

Therefore, S(1800) = 6 * 7 * 431 = 18102.

Correct answer is (c) 12

Easy explanation: To find the NUMBER of odd factors, we can exclude any POWER of 2 and do the same. So, for 6500, we have (5 + 1)(1 + 1) = 6 * 2 = 12 odd POSITIVE factors.

Right ANSWER is (a) 12493

Easy explanation: The prime FACTORIZATION of 1872 is 13 * 3^2 * 2^4 and S(2^4) = 1 + 2 + 4 + 8 + 16 = 31, S(5^2) = 1 + 5 + 25 = 31. Therefore, S(1872) = 31 * 31 * 13 = 12493.

Correct answer is (d) 8

The best explanation: To find the NUMBER of ODD FACTORS (which includes 1), we can exclude any power of 2 and do the same. So, for 456, we have (3 + 1)(1 + 1) = 8 odd positive factors.

Right choice is (a) 24

Best explanation: To find the number of EVEN factors, we can multiply the number of even factors by the POWER of 2. For 3200, we have (5 + 1)(1 + 1)(2) = 24 even factors.

Right choice is (d) 31620

To explain: The prime factorization of 1800 is 3 * 2^7 * 5^2 and

S(3) = 1 + 3 = 4

S(2^2) = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 = 255

S(5^2) = 1 + 5 + 25 = 31

Therefore, S(1800) = 4 * 255 * 31 = 31620.

Correct choice is (B) polynomial time

To explain I would SAY: The exact number of running time depends on the computational MODEL. When analyzing arithmetic with large numbers, we USUALLY count either BIT operations or arithmetic operations of size O(logn) (where n is the input size). Now, given the factorization of a number n, then the sum of divisors can be computed in polynomial time.

The correct OPTION is (d) 4123

The best explanation: The PRIME factorization of 1800 is 19 * 2^2 * 5^2 and

S(2^2) = 1 + 2 + 4 = 7

S(5^2) = 1 + 5 + 25 = 31

Therefore, S(1800) = 19 * 7 * 31 = 4123.

The CORRECT answer is (a) 686

Explanation: We MAY have (2 men and 3 WOMEN) or (1 men and 4 woman) or (5 women only). The Required number of ways = (^8C2 × ^6C3) + (^8C1 × ^6C4) + (^6C5) = 686.

The correct choice is (b) 30

For explanation I would say: Any factor of this NUMBER should be of the form 5^a * 7^b * 2^c. For the factor to be a PERFECT square a, b, c has to be even. a can take VALUES 0, 2, 4, 6, 8, b can take values 0, 2, 4 and c can take values 0, 2. Total number of perfect squares = 5 * 3 * 2 = 30.

Correct answer is (b) 480

Explanation: The total number of WAYS in which 6 part can be arranged = 6! = 720. The total number of ways in which part-1 and part-3 are ALWAYS together: = 5!*2! = 240. Therefore, the total number of arrangements, in which they are not together is = 720 − 240 = 480.

The correct choice is (d) 1317

The explanation is: The number of WAYS to choose 3 people out of 9 is ^15C3. Then, number of ways to choose 5 people out of (15-3) = 12 is ^12C5. FINALLY, the number of ways to choose 4 people out of (12-4) = 8 is ^8C4. Hence, by the RULE of product, ^15C3 + ^12C5 + ^8C4 = 1317.

The correct choice is (b) 50388

For explanation I WOULD say: Consider the situation as having 19 spots and FILLING them with 7 chocolate biscuits and 19 CHEESECAKE biscuits. Then we just choose 7 spots for the chocolate biscuits and let the other 10 spots have cheesecake biscuits. The number of ways to do this JOB is ^19C7 = 50388.

The CORRECT ANSWER is (d) ^15C4 * ^5C3 * 7!

The best explanation: There are 4 consonants out of 15 can be selected in ^15C4 ways and 3 vowels can be selected in ^5C3 ways. Therefore, the total NUMBER of groups each containing 4 consonants and 3 vowels = ^15C4 * ^4C3. Each group CONTAINS 7 letters which can be arranged in 7! ways. Hence, REQUIRED number of words = ^15C4 * ^5C3 * 7!.

Right answer is (a) 15 *12! * 2!

For explanation I would say: We know that n objects can be arranged around a circle in \(\FRAC{(n−1)!}{2}\). If we consider the TWO sisters and the person in between the brothers as a block, then there will 12 others and this block of three PEOPLE to be arranged around a circle. The number of ways of arranging 13 objects around a circle is in 12! ways. Now the sisters can be arranged on either side of the person who is in between the sisters in 2! ways. The person who sits in between the two sisters can be any of the 15 in the group and can be selected in 15 ways. Therefore, the total number of ways 15 *12! * 2!.

Right choice is (d) same AREA

The best I can explain: Since all the points are EQUALLY spaced; hence the area of all the CONVEX heptagons will be the same.

Correct option is (d) 48

The BEST I can explain: E and F can sit together in all ARRANGEMENTS in 2! Ways. Now, the arrangement of the 5 PEOPLE in a circle can be done in(5 – 1)! or 24 ways. Therefore, the total number of ways will be 24 x 2 = 48.

Right choice is (a) 6

To EXPLAIN I would say: The number of ways of selecting one ‘B’s = 1, selecting TWO ‘B’s = 1, selecting three ‘B’s = 1, selecting four ‘B’s = 1, selecting five ‘B’s = 1 and selecting six ‘B’s = 1. HENCE, the required number of ways = 6.

Correct choice is (a) 269

To elaborate: Number of ways of selecting roses = (5+1) = 6 ways, number of ways of selecting MARIGOLD = (4+1) = 5 ways, and the number of ways of selecting sunflowers = (8+1) = 9 ways. Total number of ways of selecting flowers= 6 * 5 * 9 = 270. But this includes when no flowers or zero flowers is selected (There is no flowers of a DIFFERENT TYPE, hence n=0 =>2^n= 2^0 = 1). Hence, the number of ways of selecting at least one FRUIT = 270 – 1 = 269.

The correct CHOICE is (C) 31824

Easy explanation: If a particular player is never chosen that would mean 11 PLAYERS are SELECTED out of 18 players. HENCE, required number of ways =^18C11 = 31824.

Correct option is (c) 45

To elaborate: There are six places to be filled in the multiset USING the n distinct elements. At least ONE element has to occur exactly twice and that would leave 4 more places in the multiset MEANS that at most four elements can occur exactly once. Thus there are two mutually exclusive cases as follows: 1) Exactly one element occurs exactly twice and select this element in n ways. Fill up the remaining four spots using 5 distinct elements from the remaining n−1 elements in ^n-1C4 ways. 2) Exactly four elements that occur at least once each. HENCE, the total number of ways to form the multiset is

^nC2 + n * ^n-1C4 = ^6C2 + 6 * ^6-1C4 = 45.

The correct choice is (b) 48

Best EXPLANATION: When ‘O’ and ‘A’ are occupying end-places => A.S.T.L. (CE). We can see that (CE) are fixed, HENCE A, S, T, L can be arranged in4! WAYS and (C, E) can be arranged themselves is 2! ways. So, the number of words formed =4! x2! = 48 ways.

Correct option is (B) 120

The EXPLANATION is: Here number of digits, n = 5 and number of places to be filled-up R = 3. Hence, the required number is ^5P3 = 5!/2!*3! = 120.

The correct OPTION is (B) 40320

The BEST I can explain: Now the first student is eligible to receive any of the 8 available prizes (so 8 ways), the second student will receive a prize from rest 7 available prizes (so 7 ways), the third student will receive his prize from the rest 6 prizes available(so 6 ways) and so on. So total ways would be 8! = 8*7*6*5*4*3*2*1 = 40320. Hence, the 7 prizes can be DISTRIBUTED in 40320 ways.

Right OPTION is (b) 3386880

For explanation: There are 3 SISTERS and 8 other GIRLS in total of 11 girls. The number of ways to arrange these 11 girls in a circular manner = (11– 1)! = 10!. These three sisters can now rearrange themselves in 3! ways. By the multiplication theorem, the number of ways so that 3 sisters always come TOGETHER in the arrangement = 8! × 3!. Hence, the required number of ways in which the arrangement can take place if none of the 3 sisters is seated together: 10! – (8! × 3!) = 3628800 – (40320 * 6) = 3628800 – 241920 = 3386880.

Correct choice is (c) ^19P10

Easy EXPLANATION: First let us take the 18 unoccupied seats. They create 19 SLOTS i.e., ONE on the LEFT of each seat and one on the right of the last one. So we can place the 10 boys in any of these 19 slots that are, ^19P10 WAYS.

Right answer is (b) 999988

Easy explanation: Number of WORDS which have at least one letter REPLACED = total number of words – total number of words in which no letter is REPEATED, => 10^6 – ^12P6 => 1000000 − 924 = 999988.

Correct answer is (a) 84

For EXPLANATION: Selecting any 3 elements from given 9 elements GIVES ^9C3 = 84 number of DISTINCT PAIRS of sequences.

The correct option is (b) 453600

To explain I would say: For the first position, there are 9 POSSIBLE choices (since 0 is not ALLOWED). After that number is chosen, there are 10 possible choices (since 0 is now allowed) for the SECOND digit, for the third digit there are 10 possible choices, 9 possible choices for the fourth digit and 8 possible choices for the fifth digit and 7 possible choices for the sixth digit. The COUNT of number locks = 453600.

Correct option is (a) 43758

Easiest EXPLANATION: First place 17 ZEROES side by side _ 0 _ 0 _ 0 … 0 _ and 8 1’s can be placed in any of the (17+1) AVAILABLE GAPS hence the number of ways= ^n+1Ck = 43758.

Correct answer is (c) 729

Easiest explanation: The word ‘SWIMMING contains 8 letters. Of which, I occurs twice and M occurs twice. Therefore, the number of words formed by this word = \(\frac{8!}{2!*2!}\) = 10080. In ORDER to find the number of permutations that can be formed where the two vowels I and I come together,we group the letters that should come together and consider that group as one letter. So, the letters are S, W, M, M, N, G, (I, I). So, the number of letters are 7 the number of WAYS in which 7 letters can be arranged is 7! = 5040. In I and I, the number of ways in which I and I can be arranged is 2!. Hence, the total number of ways in which the letters of the ‘SWIMMING’ can be arranged such that vowels are always together are \(\frac{7!}{2!*2!}\) = 5040 ways. The number of words in which the vowels do not come together is = (10080 – 5040) = 5040.