Permutation And Combination - Study Mode

[#126] After every get-together every person present shakes the hand of every other person. If there were 105 handshakes in all, how many persons were present in the party?
Correct Answer

(B) 15

Explanation

Solution: Let total number of persons present in the party be x,
Then, $$eqalign{
& frac{{x imes left( {x - 1}
ight)}}{2} = 105 cr
& x = 15 cr} $$

[#127] How many diagonals can be drawn in a pentagon?
Correct Answer

(A) 5

Explanation

Solution: A pentagon has 5 sides. We obtain the diagonals by joining the vertices in pairs. Total number of sides and diagonals, = 5 C 2 = $$frac{{5 imes 4}}{{2 imes 1}}$$ = 5 × 2 = 10 This includes its 5 sides also. ⇒ Diagonals = 10 – 5 = 5 Hence, the number of diagonals = 10 – 5 = 5

[#128] 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?
Correct Answer

(A) 242

Explanation

Solution: 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 = 120 + 120 + 2 = 242 Note: 5! = 5 × 4 × 3 × 2 × 1 = 120

[#129] A class photograph has to be taken. The front row consists of 6 girls who are sitting. 20 boys are standing behind. The two corner positions are reserved for the 2 tallest boys. In how many ways can the students be arranged?
Correct Answer

(B) 18! × 1440

Explanation

Solution: Two tallest boys can be arranged in 2! ways, rest 18 can be arranged in 18! ways. Girls can be arranged in 6! ways. Total number of ways in which all the students can be arranged, = 2! × 18! × 6! = 18! × 1440 Note: N! = N × (N - 1) × (N - 2) × . . . . × 1 So, 18! = 18 × 17 × 16 × 15 . . . . . . . . × 1

[#130] If $$5{ imes ^{ ext{n}}}{{ ext{P}}_3} = 4{ imes ^{left( {{ ext{n}} + 1}
ight)}}{{ ext{P}}_{3,}}$$ xa0 xa0find n?
Correct Answer

(D) 14

Explanation

Solution: n P 3 = n × (n–1) × (n–2) (n+1) P 3 = (n+1) × n × (n–1) Now, 5 × n × (n–1) × (n–2) = 4 × (n+1) × n × (n–1) Or, 5(n−2) = 4(n+1) Or, 5n − 10 = 4n + 4 Or, 5n − 4n = 4 + 10 Hence, n = 14 Note: n P r = $$frac{{{ ext{n}}!}}{{left( {{ ext{n}} - { ext{r}}}
ight)!}}$$