Permutation and Combination
1.In how many ways can 8 Indians and, 4 American and 4 Englishmen can be seated in a row so that all person of the same nationality sit together?

Explanation:

Solution:
Taking all person of same nationality as one person, then we will have only three people.
These three person can be arranged themselves in 3! Ways.
8 Indians can be arranged themselves in 8! Way.
4 American can be arranged themselves in 4! Ways.
4 Englishman can be arranged themselves in 4! Ways.
Hence, required number of ways = 3!*8!*4!*4! Ways.

2.How many Permutations of the letters of the word APPLE are there?

Explanation:

Solution:
APPLE = 5 letters.
But two letters PP is of same kind.
Thus, required permutations,
= 5!/2!
= 120/2
= 60.

3.How many different words can be formed using all the letters of the word ALLAHABAD?
(a) When vowels occupy the even positions.
(b) Both L do not occur together.

Explanation:

Solution:
ALLAHABAD = 9 letters. Out of these 9 letters there is 4 A's and 2 L's are there.
So, permutations = 9!/4!.2!= 7560.

(a) There are 4 vowels and all are alike i.e. 4A's.
_2nd _4th _6th _8th _
These even places can be occupied by 4 vowels. In 4!/4! = 1 Way.
In other five places 5 other letter can be occupied of which two are alike i.e. 2L's.
Number of ways = 5!/2! Ways.
Hence, total number of ways in which vowels occupy the even places = 5!/2! *1 = 60 ways.

(b) Taking both L's together and treating them as one letter we have 8 letters out of which A repeats 4 times and others are distinct.
These 8 letters can be arranged in 8!/4! = 1680 ways.
Also two L can be arranged themselves in 2! ways.
So, Total no. of ways in which L are together = 1680 * 2 = 3360 ways.
Now,
Total arrangement in which L never occur together,
= Total arrangement - Total no. of ways in which L occur together.
= 7560 - 3360 = 4200 ways.

4.In how many ways 4 boys and 3 girls can be seated in a row so that they are alternate.

Explanation:

Solution:
Let the Arrangement be,
B G B G B G B
4 boys can be seated in 4! Ways.
Girl can be seated in 3! Ways.
Required number of ways,
= 4!*3! = 144.

5.A two member committee comprising of one male and one female member is to be constitute out of five males and three females. Amongst the females. Ms. A refuses to be a member of the committee in which Mr. B is taken as the member. In how many different ways can the committee be constituted ?

Explanation:

Solution:
5C1 * 3C1 -1
= 15-1
= 14

6.In how many ways 2 students can be chosen from the class of 20 students?

Explanation:

Solution:
Number of ways = 20C2 = 20!/(2! 18!)
= 20*19/2
= 190.

7.Three gentlemen and three ladies are candidates for two vacancies. A voter has to vote for two candidates. In how many ways can one cast his vote?

Explanation:

Solution:
There are 6 candidates and a voter has to vote for any two of them.
So, the required number of ways is,
= 6C2 = 6!/2!*4!
= 15.

8.Find the number of triangles which can be formed by joining the angular points of a polygon of 8 sides as vertices.

Explanation:

Solution:
A triangle needs 3 points.
And polygon of 8 sides has 8 angular points.
Hence, number of triangle formed,
= 8C3 = (8*7*6)/(1*2*3)
= 56.

9.There are 10 points in a plane out of which 4 are collinear. Find the number of triangles formed by the points as vertices.

Explanation:

Solution:
The number of triangle can be formed by 10 points = 10C3.
Similarly, the number of triangle can be formed by 4 points when no one is collinear=4C3.
In the question, given 4 points are collinear, Thus, required number of triangle can be formed,
= 10C3-4C3 = 120-4 = 116.

10. 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.

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 nC2.

Now, according to the question,

nC2 = 105;
Or, n!/[2!*(n-2)!] = 105;
Or, n*(n-1)/2 = 105;
Or, n2-n = 210;
Or, n2-n-210 = 0;
Or, n = 15, -14;

But, we cannot take negative value of n.
So, n = 15 i.e. number of persons in the party = 15.

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