Permutations and Combinations jee mains questions | JEE Main Maths Permutations and Combinations Previous Year Question

Q 11 :
The total number of words (with or without meaning) that can be formed out of the letters of the word 'DISTRIBUTION' taken four at a time, is equal to _______. [2024]

0%

(3734)

The word 'DISTRIBUTION' has total 12 letters and 9 diferent letters.

We select four different letters.

Case l : Three same letters and 1 different letter ${}^{1}{C_{1}} \times {}^{8}{C_{1}} \times \frac{4!}{3!}$

Case 2 :Two pairs are same and two pairs are different ${}^{2}{C_{1}} \times {}^{8}{C_{2}} \times \frac{4!}{2!}$

Case 3 : Two pair of letters are same ${}^{2}{C_{2}} \cdot {}^{4}{C_{2}}$

Case 4 : All different letters are select ${}^{9}{C_{4}} \times 4!$

Total letters selected.

$= {}^{1}{C_{1}} \times {}^{8}{C_{1}} \times \frac{4!}{3!} + {}^{2}{C_{1}} \times {}^{8}{C_{2}} \cdot \frac{4!}{2!} + {}^{2}{C_{2}} \cdot {}^{4}{C_{2}} + {}^{9}{C_{4}} \cdot 4!$

$= 1 \times \frac{8!}{7!} \times \frac{4!}{3!} + 2! \times \frac{8!}{6! 2!} \times \frac{4!}{2!} + \frac{1 \times 4!}{2! 2!} + \frac{9! \times 4!}{5! 4!}$

$= 32 + 672 + 6 + 3024 = 3734$

Q 12 :
There are 5 points $P_{1}, P_{2}, P_{3}, P_{4}, P_{5}$ on the side AB, excluding A and B, of a triangle ABC. Similarly, there are 6 points $P_{6}, P_{7}, \dots, P_{11}$ on the side BC and 7 points $P_{12}, P_{13}, \dots, P_{18}$ on the side CA of the triangle. The number of triangles, that can be formed using the points $P_{1}, P_{2}, \dots, P_{18}$ as vertices, is: [2024]

796
751
771
776

1

2

3

4

(2)

Number of points on side AB = 5
Number of points on side BC = 6
Number of points on side AC = 7

Number of ways of selecting three points from side AB = ${}^{5}{C_{3}} $

Number of ways of selecting three points from side BC = ${}^{6}{C_{3}} $

Number of ways of selecting three points from side AC = ${}^{7}{C_{3}}$

Total number of triangles that can be formed using the points $P_{1}, P_{2}, \dots, P_{18} = {}^{18}{C_{3}} - ({}^{5}{C_{3} + {}^{6}{C_{3} + {}^{7}{C_{3}}}})$

$= {}^{18}{C_{3} - {}^{5}{C_{3}} - {}^{6}{C_{3}} - {}^{7}C_{3}} = 816 - 10 - 20 - 35 = 751$

Q 13 :
The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is: [2024]

16
48
24
56

1

2

3

4

(1)

Total number of triangles that can be formed by using 8 vertices of the octagon = ${}^{8}{C_{3}}$

Number of triangles having exactly one side common with the octagon = $8 \times 4$

Since, if we choose AB as the common side, then other vertices of the triangle will be either of G, F, E or D as we have exactly one common side.

So, we have $8 \times 4$ i.e., $32$ such triangles.

Now, let us find the number of triangles having two common sides.

In the octagon, we have 8 ways of choosing two consecutive sides, i.e., (AB, BC), (BC, CD), (CD, DE), (DE, EF), (EF, FG), (FG, GH), (GH, HA), (HA, AB)

$∴$ Number of triangles having 2 sides common with the octagon = 8

$∴$ Required number of triangles = ${}^{8}{C_{3}} - 32 - 8$

$= 56 - 32 - 8 = 16$

Q 14 :
If for some $m, n;$ ${}^{6}{C_{m}} + 2 ({}^{6}{C_{m + 1}}) +^{6} C_{m + 2} >^{8} C_{3}$ and ${}^{n - 1}{P_{3}} :^{n} P_{4} = 1 : 8,$ then ${}^{n}{P_{m + 1}} +^{n + 1} C_{m}$ is equal to [2024]

372
384
380
376

1

2

3

4

(1)

Given, ${}^{6}{C_{m}} + 2 ({}^{6}{C_{m + 1}}) +^{6} C_{m + 2} >^{8} C_{3}$

$\Rightarrow^{6} C_{m} +^{6} C_{m + 1} +^{6} C_{m + 1} +^{6} C_{m + 2} >^{8} C_{3}$

$\Rightarrow^{7} C_{m + 1} +^{7} C_{m + 2} >^{8} C_{3} \Rightarrow^{8} C_{m + 2} >^{8} C_{3} {or}^{8} C_{6 - m} >^{8} C_{3}$

$\Rightarrow m + 2 > 3$ i.e., $m > 1$

or $6 - m > 3$ i.e., $m < 3$ i.e., $1 < m < 3 \Rightarrow m = 2$

$∴$ ${}^{n}{P_{m + 1}} +^{n + 1} C_{m} =^{8} P_{3} +^{9} C_{2} = 336 + 36 = 372$

Topic Question Set

Q 11 :
The total number of words (with or without meaning) that can be formed out of the letters of the word 'DISTRIBUTION' taken four at a time, is equal to _______. [2024]

0%

Q 13 :
The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is: [2024]

Q 14 :
If for some $m, n;$ ${}^{6}{C_{m}} + 2 ({}^{6}{C_{m + 1}}) +^{6} C_{m + 2} >^{8} C_{3}$ and ${}^{n - 1}{P_{3}} :^{n} P_{4} = 1 : 8,$ then ${}^{n}{P_{m + 1}} +^{n + 1} C_{m}$ is equal to [2024]

Want Us to Email you About Special Offers & Updates?

Topic Question Set

Q 11 : The total number of words (with or without meaning) that can be formed out of the letters of the word 'DISTRIBUTION' taken four at a time, is equal to _______. [2024]

0%

Q 13 : The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is: [2024]

Q 14 : If for some m,n; Cm6+2(Cm+16)+6Cm+2>8C3 and P3n-1:nP4=1:8, then Pm+1 n+ n+1Cm is equal to [2024]

Want Us to Email you About Special Offers & Updates?

Q 11 :
The total number of words (with or without meaning) that can be formed out of the letters of the word 'DISTRIBUTION' taken four at a time, is equal to _______. [2024]

Q 13 :
The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is: [2024]

Q 14 :
If for some $m, n;$ ${}^{6}{C_{m}} + 2 ({}^{6}{C_{m + 1}}) +^{6} C_{m + 2} >^{8} C_{3}$ and ${}^{n - 1}{P_{3}} :^{n} P_{4} = 1 : 8,$ then ${}^{n}{P_{m + 1}} +^{n + 1} C_{m}$ is equal to [2024]