Permutation And Combination - Study Mode

[#146] Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed?
Correct Answer

(C) 25200

Explanation

Solution: Number of ways of selecting (3 consonants out of 7) and (2 vowels out of 4) $$eqalign{
& = left( {{}^7{C_3} imes {}^4{C_2}}
ight) cr
& = left( {frac{{7 imes 6 imes 5}}{{3 imes 2 imes 1}} imes frac{{4 imes 3}}{{2 imes 1}}}
ight) cr
& = 210 cr} $$ Number of groups, each having 3 consonants and 2 vowels = 210 Each group contains 5 letters. Number of ways of arranging 5 letters among themselves = 5! = 5 x 4 x 3 x 2 x 1 = 120 ∴ Required number of ways = (210 x 120) = 25200

[#147] In how many ways can the letters of the word 'LEADER' be arranged?
Correct Answer

(C) 360

Explanation

Solution: The word 'LEADER' contains 6 letters, namely 1L, 2E, 1A, 1D and 1R. ∴ Required number of ways
$$ = frac{{6!}}{{left( {1!}
ight)left( {2!}
ight)left( {1!}
ight)left( {1!}
ight)left( {1!}
ight)}} = 360$$

[#148] There are 10 points in a plane out of which 4 are collinear. Find the number of triangles formed by the points as vertices.
Correct Answer

(B) 116

Explanation

Solution: The number of triangle can be formed by 10 points = 10 C 3 Similarly, the number of triangle can be formed by 4 points when no one is collinear = 4 C 3 In the question, given 4 points are collinear,
Thus, required number of triangle can be formed, = 10 C 3 - 4 C 3 = 120 - 4 = 116

[#149] In a party every person shakes hands with every other person. If there are 105 hands shakes, find the number of person in the party.
Correct Answer

(A) 15

Explanation

Solution: Let n be the number of persons in the party Number of hands shake = 105 Total number of hands shake is given by n C 2 Now, According to the question,
$$eqalign{
& ^n{{ ext{C}}_2} = 105 cr
& { ext{or, }}frac{{n!}}{{2! imes left( {n - 2}
ight)!}} = 105 cr
& { ext{or, }}frac{{n imes left( {n - 1}
ight)}}{2} = 105 cr
& { ext{or, }}{n^2} - n = 210 cr
& { ext{or, }}{n^2} - n - 210 = 0 cr
& { ext{or, }}n = 15,, - 14 cr} $$
But, we cannot take negative value of n
So, n = 15 i.e. number of persons in the party = 15

[#150] The number of positive integers which can be formed by using any number of digits from 0, 1, 2, 3, 4, 5 without repetition.
Correct Answer

(D) 1630

Explanation

Solution: One digit positive numbers = 5
Two digit positive numbers = 25
Three digit positive numbers = 100 4 digit positive numbers = 300 5 digit positive numbers = 600 Six digit positive numbers = 600 Total positive numbers, = 5 + 25 + 100 + 300 + 600 + 600 = 1630