CBSE Class 11 Mathematics Study Material All Chapters

CHAPTER-7

(PERMUTATION & COMBINATION)

Points to be remembered

1. Fundamental principle of counting:-

If one operation can be performed in 'm' different ways and if corresponding to each way of performing this operation,there are 'n' different ways of performing second operation ,the two operations together can be performed in (mXn) ways.

2. Factorial:-

The continued product of first 'n' natural numbers is called factorial 'n' or 'n' factorial and is denoted as 'n!'.

3. 0! has no meaning. Conventionally, we define 0! =1.

4. The number of permutation (arrangements of 'n' different things taken 'r' at a time given by ⁿP_r (0<=r<=n)
ⁿP_r = n!/(n-r)!

5. The number of ways in which 'r' vacant places can be filled up with 'n' different things is given by nr.

6. The number of permutations of 'n' things taken altogether,when 'p' of the things are alike of one kind, 'q' of them alike of another kind, 'r' of them alike of a third kind and the remaining all different is given by:- n!/(p!q!r!).

7. The number of ways of arranging 'n' distinct objects along a round table=(n-1)!

8. The number of ways of arranging 'n' persons along a round table so that no person has the same two neighbours=(n-1)!/2

9. In forming a necklace, there is no distinction between a clockwise and anticlock wise direction. Hence, the no. of necklace formed with 'n' beads of different colours =(n-1)!/2

10. The number of combination of 'n' different things taken 'r' at a time is given by nCr, (0<=r<=n)

nCr= n!/r!(n-r)!

11. ⁿC_r + ⁿC_(r-1) =⁽ⁿ⁺¹⁾C_r where 1<=r<=n, and n,r belongs to N.

12. ⁿC_r/ⁿC_(r-1) = (n-r+1)/r where n,r belongs to N, 1<=r<=n.

13. The number of ways in which (m+n) things can be divided into two groups containing 'm' and 'n' things respectively is given by= (m+n)!/m! n!

SHORT ANSWER TYPE QUESTIONS

1. Find HCF & LCM of 4!,5!,6!?

2. If 1/8! + 3/7! = x/9!, find x?

3. Find 'n' if 5 P(4,n)=6 P(5,n-1)?

4. In how many ways can the letters of the word "EQUATION" be arranged?

5. Ten students are participating in a race. In how many ways can the first three prizes be won?

6. In how many ways can 4 books on mathematics and 3 books on English be placed on a shelf so that books on the same subject always remain.

7. How many words can be formed out of letters of the word "TRIANGLE". How many of those will begin with 'T' and end with 'E'?

8. In how many ways can 3 prizes be given to 10 students when a student can receive any number of prizes?

9. How many different arrangements can be made from the letters of the word "ENGINEERING"?

10. In how many ways can 6 beads of different colours form a necklace?

11. Evaluate ¹⁰C₇ + ¹⁰C₆ ?

12. If C(2n,3):C(n,3):: 11:1 ,find 'n'?

13. Find the no. of straight lines and no. of triangles that can be formed from 15 non collinear points?

14. Find the no. of all 4-digit nos. with no digit being repeated?

15. Find the no. of all 3-digit even natural nos?

LONG ANSWER TYPE QUESTIONS(4 MARKS)

1. How many nos. greater than 56000 can be formed by using the digits 4,5,6,7,8 if no digit is repeated in any no.?

2. Find the no. of ways in which the letters of "ARRANGEMENT" can be arranged so that the 2 R's and 2 A's don't occur together?

3. A committee of 6 is to be formed from 6 boys and 4 girls. In how many ways can this be done if the committee contains :
(a) 2 girls?
(b) atleast 2 girls?

4. A man has 7 relatives , 4 of them are ladies and 3 gentlemen, his wife also has 7 relatives , 3 of them are ladies and 4 gentlement. In how many ways can they invite a dinner party of 3 ladies and 3 gentleman so that there are 3 of man's relatives and 3 of wife's relatives?

5. Find the no. of all possible arrangement of the letters of the word "MATHEMATICS" taken 4 at a time?

6. From a class of 25 students, 10 are to be chosen for an excursion party. there are 3 students who decide that either all of them will join or none of them will join. In how many ways can the excursion party be chosen?

7. Find the no. of ways in which 5 boys and 5 girls can be seated in 10 seats in a row if the boys and girls are to be seated alternately?

8. In how many ways can the letters of the word "ASSASSINATION" be arranged so that all the S's are together?

9. In an experiment, a question paper consists of 12 questions divided into 2 parts i.e part1 & part2 containing 5 & 7 questions. A student is required to attempt 8 questions in all, selecting atleast 3 from each part. In how many ways can a student select the questions?

10. How many 6-digit nos. can be formed from the digits 0,1,3,5,7,9 which are divisible by 10 and no digit is repeated?

11. Prove that ²ⁿC_n= (1.3.5.7.....(2n-1)X2n)/n!

12. How many even nos. are there with 3-digits such that if 5 is one of the digits, then '7' is the next digit?

13.A teacher wants to arrange 5 students on the plat form such that the buy YUSUF occupies the first position and the girls GEETA & SEETA are always together. How many such arrangements are possible?

14. How many different nos. with distinct digits can be formed from the digits 2,3,5,7,9. How many of them are odd?

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