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Applications Of Selections - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

  • APPLICATIONS OF SELECTIONS -II is considered one the most difficult concept.

  • 294 Questions around this concept.

Solve by difficulty

Determine the number of ways two numbers from the prime number set upto 200 and multiplied together to obtain a product that is a multiple of neither 5 nor 3.

 

Calculate the number of ways 2 numbers can be chosen from the set containing perfect square positive integers till 111 and multiplied together to obtain a product that is a multiple of 4 or 6.

 

A password must be 8 characters long and include exactly 3 uppercase letters, 4 lowercase letters, and 1 digits. How many different passwords are possible?

 

A chess tournament has 8 players. In how many different ways can the players be paired for the first round?

 

A restaurant offers a menu with 4 appetizers, 5 main courses, and 6 desserts. In how many ways can a customer choose one appetizer, one main course, and one dessert?

 

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A restaurant offers a menu with 4 appetizers, 10 main courses, and 6 desserts. In how many ways can a customer choose two appetizers, one main course, and one dessert?

A restaurant offers a menu with 4 appetizers, 10 main courses, and 6 desserts. In how many ways can a customer choose one appetizer, two main courses, and one dessert?

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A pizza place offers 5 different toppings, but a customer must choose at least 2 toppings for their pizza. How many different pizza combinations are possible?

The company has 7 job openings and 18 qualified candidates. Each candidate can only be selected for one position. In how many ways can you choose a combination of candidates to fill the job positions?

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The company has 6 job openings and 10 qualified candidates. Each candidate can only be selected for one position. In how many ways can you choose a combination of candidates to fill the job positions?

Concepts Covered - 4

APPLICATIONS OF SELECTIONS -I

Let us take an example of selecting things from two or more different groups:

Out of 5 men and 6 women in how many ways can a committee of 5 members be selected such that at least 2 members are women? 

Solution:

Following cases are possible for at least 2 women,

2 women + 3 men = \\\mathrm{^6C_2\times ^5C_3}

3 women + 2 men = \\\mathrm{^6C_3\times ^5C_2}

4 women + 1 men = \\\mathrm{^6C_4\times ^5C_1}

5 women = \\\mathrm{^6C_5}

So, the total number of ways \\\mathrm{=^6C_2\times ^5C_3+^6C_3\times ^5C_2 + ^6C_4\times ^5C_1 + ^6C_5 = 431}

APPLICATIONS OF SELECTIONS -II

Restricted Combination

The number of selection of r objects from n different objects:

  1. When k particular things are always included  = \mathrm{^{n-k}C_{r-k}}

This can be comprehended as taking out those k things which have to be included which can be done in 1 way and then finding the ways in which r-k objects can be selected from remaining n - k things, and putting those k things (which are already taken out) in r-k selected objects.

  1. k particular things are never included  = \mathrm{^{n-k}C_{r}}

This can be comprehended as taking out k things which are not to be selected which can be done in 1 way and then finding the ways of selecting r things from n-k things.

  1. The number of ways selecting r things out of n different things such that p particular objects are always included and q particular objects are always excluded = \mathrm{^{n-p-q}C_{r-p}}

This can be comprehended as taking out the q objects which should not be selected and putting it out and then taking out p objects which have to be selected and then finding ways of selecting r-p objects out of n- p-q objects and putting back p objects in r-p selected objects.

 

Example: In how many ways a cricket team can be selected out of 16 players such that 5 certain players must be included in the team.

Solution: Since 5 certain player has to be included so be need to select 11-5 = 6 player from 16 - 5 = 11 player.

      So we can select the team in \\\mathrm{^{16-5}C_{11-5}=^{11}C_6=\frac{11!}{5!6!}}

APPLICATIONS OF SELECTIONS - III (GEOMETRICAL APPLICATIONS)

If there are n points in the plane  and out of which no three are collinear then,

  1. Total No. of lines that can be formed using these n points = nC2

  2. Total No. of triangles that can be formed using these n points = nC3

  3. Total no. of Diagonals that can be formed in n sided polygon = nC2 - n

 

If there are n points in the plane and out of which m points are collinear, then,

  1. Total No. of different lines that can be formed by joining these n points is \mathrm{^nC_2-^mC_2+1}

  1. Total No. of different triangles that can be formed by joining these n points is \mathrm{^nC_3-^mC_3}

  1. Total No. of different quadrilaterals formed by joining these n points is \mathrm{^nC_4-\left (^mC_3 \cdot^{n-m}C_1+^mC_4\right )}

 

Number of Parallelograms

If m parallel lines in a plane are intersected by the family of other n parallel lines, then the total number of parallelograms formed is 

\mathrm{^mC_2\cdot ^nC_2=\frac{mn(m-1)(n-1)}{4}}

 

Number of rectangles and squares

  1. Number of rectangles of any size in a square of size n x n is \sum_{r=1}^{n}r^3 and number of squares of any size is \sum_{r=1}^{n}r^2.

  2. In a rectangle of size n x p (n < p) number of rectangles of any size is \frac{np}{4}(n+1)(p+1).

APPLICATIONS OF SELECTIONS-IV

To determine the number of ways to reach in the shortest way from point A to B.

When considering the possible paths or shortest path one can observe that the total number of steps in the forward direction is 6-R(Right) and in the upward direction is 4-U(Upward)

Now, If we arrange these 6 Rs and 4 Us in any way, it comes out to be a shortest path.

Or one can say that first find all the possible steps and arrange them to get the total number of possible ways.

Using "u" and "r" we can write out a path:

r r r r r r u u u u

r r r u u u u r r r

and others......

Hence, the total number of ways is \frac{10!}{4!6!} or, ^{10}C_4

Study it with Videos

APPLICATIONS OF SELECTIONS -I
APPLICATIONS OF SELECTIONS -II
APPLICATIONS OF SELECTIONS - III (GEOMETRICAL APPLICATIONS)

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Books

Reference Books

APPLICATIONS OF SELECTIONS -I

Mathematics Textbook for Class XI

Page No. : 150

Line : 11

APPLICATIONS OF SELECTIONS -II

Algebra (Arihant)

Page No. : 376

Line : 28

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