The letter of the word LABOUR are permuted in all possible ways and the words thus formed are arranged as in a dictionary. What is the rank of the word LABOUR?

Answer: D The order of each letter in the dictionary is ABLORU. Now, with A in the beginning, the remaining letters can be permuted in 5! ways. Similarly, with B in the beginning, the remaining letters can be permuted in 5! ways. With L in the beginning, the first word will be LABORU, the second will be LABOUR. Hence, the rank of the word LABOUR is 5!+5!+2 = 242.

Q. No. 2:

From 6 men and 4 ladies, a committee of 5 is to be formed. In how many ways can this be done, if the committee is to include at least one lady?

Answer: A To committee can be formed in the following ways 1 lady + 4 gents or 2 ladies + 3 gents or 3 ladies + 2 gents or 4 ladies + 1 gent or 5 ladies + 0 gent. Total number of possible arrangements:- (4_{C1}*6_{C4}) + (4_{C2}*6_{C3}) + (4_{C3}*6_{C2}) + (4_{C4}*6_{C1}) => 60+120+60+6 = 246

Q. No. 3:

In a railway compartment, there are 2 rows of seats facing each other with accommodation for 5 in each, 4 wish to sit facing forward and 3 facing towards the rear while 3 others are indifferent. In how many ways can the 10 passengers be seated?

Answer: D The four person who wish to sit facing forward can be seated in 5_{P4} ways and 3 who wish to sit facing towards the rear can be seated in 5_{P3} ways and the remaining 3 can be seated in the remaining 3 seats in 3_{P3} ways. Total number of ways = 5_{P4}*5_{P3}*3_{P3} = 43200

Q. No. 4:

From 3 mangoes, 4 apples and 2 oranges, how many selections of fruits can be made, taking at least one of each kind?

Answer: C Number of ways in which mangoes can be selected =2^{3}. But this also includes, the case where all three mangoes are not selected. Hence, number of ways in which at least mango is selected = (2^{3}-1). Similarly, number of ways in which apples and oranges are selected is (2^{4}-1) and (2^{2}-1). Hence, total number of selections = (2^{3}-1)*(2^{4}-1)*(2^{2}-1) = 315

Q. No. 5:

Find the number of ways in which 21 books may be arranged on a shelf so that the oldest and the newest books never come together.

Answer: C The number of ways in which 21 books can be arranged = 21!. Now if the oldest and the newest book are to come together, then considering the books as one, 20 books can be arranged in 20! ways. But these two books can be arranged among themselves is 2! ways. Hence, Number of arrangements that oldest and newest books do not come together = 21! - 2*20! = 20!(21-2) = 19*20!.

Q. No. 6:

There 5 boys and 3 girls. In how many ways can they be seated in a row so that all the three girls do not sit together?

Answer: D When there is no restriction, 8 person can be seated in 8! ways. But when all the three girls sit together, consider three girls as a one, we have only 5+1= 6 persons. These 6 persons can be seated in 6! ways. But these three girls can be arranged among themselves in 3! ways. Required number of ways in which all the three girls do not sit together => 8! - (6!*3!) = 6!(56-6)= 50 * 6!.