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1. Six X's have to be placed in the squares of figure below in such a way that each row contains at least one X. In how many different ways can this be done. (1978)

Subjective Type Questions: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Ans. Sol.  As all the X’s are identical, the question is of selection of 6 squares from 8 squares, so that no row remains empty. Here R1 has 2 squares, R2 has 4 squares and R3 has 2 squares.
The selection scheme is as follows :

Subjective Type Questions: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

∴ Number of selections are

Subjective Type Questions: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

 

 

Q.2. Five balls of different colours are to be placed in there boxes of different size. Each box can hold all five. In how many different ways can we place the balls so that no box remains empty ?

Ans. 

Sol.  The various possibilities to put 5 different balls in 3 different size boxes, when no box remains empty : The balls can be 1, 1 and 3 in different boxes or 2, 2, 1.
Case I : To put 1, 1 and 3 balls in different boxes. Selection of 1, 1 and 3 balls out of 5 balls  can be done in 5C1 × 4C1 × 3Cways and then 1, 1, 3 can permute (as defferent size boxes) in 3! ways.
∴ No. of ways =  5C1 × 4C1 × 3C3× 3! = 5 × 4× 1 × 6 = 120

Case II : To put 2, 2 and 1 ball in different boxes. Selection of 2, 2 and 1 balls out of 5 balls can be done in 5C× 3C× 1Cways and then 2, 2, 1 can permute (different boxes) in 3! ways
∴ No. of ways  =  5C1 × 3C2 × 1C1× 3! = 10 × 3 × 1 × 6 = 180
Combining case I and II, total number of required  ways are  = 120 + 180 = 300.

 

 

Q.3. m men and n women are to be seated in a row so that no two women sit together. If m > n, then show that the number of ways in which they can be seated is

Subjective Type Questions: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE(1983 - 2 Marks) 

Ans.

 Sol.  m men can be seated in m! ways creating (m +1) places for ladies to sit. n ladies out of (m + 1) places (as n < m) can be seated in m + 1Pn ways
∴ Total ways = m! × m + 1Pn

Subjective Type Questions: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

 

 

Q.4. 7 relatives of a man comprises 4 ladies and 3 gentlemen; his wife has also 7 relatives; 3 of them are ladies and 4 gentlemen. In how many ways  can they invite a dinner party of 3 ladies and 3 gentlemen so that there are 3 of man’s relatives and 3 of the wife’s relatives? (1985 - 5 Marks)

Ans. 

Sol.  There are four possibilities:
(i) 3 ladies from husband’s side and 3 gentlemen from wife’s side.
No. of ways in this case = 4C3 × 4C3 = 4 × 4 = 16

(ii) 3 gentlemen from husband’s side and 3 ladies from wife’s side.
No. of ways in this case= 3C3 × 3C= 1 × 1 = 1

(iii) 2 ladies and one gentlemen from husband’s side and one lady and 2 gentlemen from wife’s side.
No. of ways in this case = (4C2 × 3C1 ) × (3C× 4C2) = 6 × 3 × 3× 6 = 324

(iv) One lady and 2 gentlemen from husband’s side and 2 ladies and one gentlemen from wife’s side.
No. of ways in this case = (4C1 × 3C2 ) × (3C2 × 4C1 ) = 4 × 3 × 3× 4 = 144
Hence the total no. of ways are = 16 + 1 + 324 + 144 = 485

 

 

Q.5. A box contains two white balls, three black balls and four red balls. In how many ways can three balls be drawn from the box if at least one black ball is to be included in the draw? (1986 - 2½ Marks)

Ans. 

Sol.  Number of ways of drawing at least one black ball
= 1 black and 2 other or 2 black and 1 other or 3 black
= 3C1 × 6C2  + 3C2 × 6C1 + 3C3
= 3 × 15 + 3 × 6+1
= 45 + 18 + 1 = 64

 

Q.6. Eighteen guests have to be seated, half on each side of a long table. Four particular guests desire to sit on one particular side and three others on the other side. Determine the number of ways in which the sitting arrangements can be made. (1991 -  4 Marks)

Ans. 

Sol.  Out of 18 guests half  i.e. 9 to be seated on side A and rest 9 on side B. Now out of 18 guests, 4 particular guests desire to sit on one particular side say side A and other 3 on other side B. Out of rest 18 – 4 – 3 = 11 guests we can select 5  more for side A and rest 6 can be seated on side  B. Selection of 5 out of 11 can be done in 11C5 ways and 9 guests on each sides of table can be seated in 9! × 9! ways. Thus there are total 11C5 × 9! × 9! arrangements.

 

Q.7. A committee of 12 is to be formed from 9 women and 8 men.
 In how many ways this can be done if at least five women have to be included in a committee? In how many of these committees (1994 -  4 Marks) (a) The women are in majority? (b) The men are in majority?

Ans. 

Sol.  Given that there are 9 women and 8 men. A committee of 12 is to be formed including at least 5 women.
This can be doen in the following ways.
=5W and 7M or 6W and  6M
or 7W and  5M
or 8W and  4M
or 9W and  3M
No. of ways of forming committee is

= 9C5 × 8C7  + 9C6 × 8C6 + 9C7 × 8C5 + 9C8 ×  8C4 + 9C9 × 8C3

Subjective Type Questions: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEESubjective Type Questions: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

= 126 × 8 + 84 × 28 + 36 × 56 + 9× 70 + 56 = 6062 ways.

(a) The women are in majority in 2016 + 630 + 56 = 2702 ways.

(b) The men are in majority in 1008 ways.

 

 

Q.8. Prove by permutation or oth erwise  Subjective Type Questions: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE  is an integer (n ∈ I+). (2004 - 2 Marks)

Ans.

Sol.  Let there be n sets of different objects each set containing n identical objects [eg (1, 1, 1 ... 1 (n times) ), (2, 2, 2 ..., 2
(n times) ... (n, n, n ... n (n times))]
Then the no. of ways in which these n × n = nobjects can be arranged in a row =Subjective Type Questions: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

But these number of ways should be a natural number.

Hence  Subjective Type Questions: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEis an integer. (n ∈ I+)

 

Q.9. If total number of runs scored in n matches is

Subjective Type Questions: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE where n > 1, and the runs scored in the kth match are given by k. 2n + 1– k, where 1 ≤ k ≤ n. Find n.             (2005 - 2 Marks)

Ans. Sol.  Given that runs scored in kth match  Subjective Type Questions: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

and runs scored in n matches  =Subjective Type Questions: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

∴ Subjective Type Questions: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE              ....(i)

Let        Subjective Type Questions: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subtracting the above two, we get

Subjective Type Questions: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE⇒  Subjective Type Questions: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

∴ Equation (i) becomes

Subjective Type Questions: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEESubjective Type Questions: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE  Subjective Type Questions: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Permutations and Combinations | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

 

 

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