Class name Subject Book title Chapter number Content Created at
Class XI Mathematics Mathematics 6 )( ) or (6 – r) (5 – r) = 6 or r – 11r + 24 = 0 or r – 8r – 3r + 24 = 0 or (r – 8) (r – 3) = 0 or r = 8 or r = 3. Hence r = 8, 3. Example 14 Find t...
Class XI Mathematics Mathematics 6 vowels occur together. Reprint 2026-27 114 MATHEMATICS = 1663200 – 16800 = 1646400 (iv) Let us fix I and P at the extreme ends (I at the left end a...
Class XI Mathematics Mathematics 6 and S? 6.4 Combinations Let us now assume that there is a group of 3 lawn tennis players X, Y, Z. A team consisting of 2 players is to be formed. I...
Class XI Mathematics Mathematics 6 Therefore P = C × 2! or = C4 2 2 2 (4 2 !)! Now, let us suppose that we have 5 different objectsA, B, C, D, E. Taking 3 at a time, if we have to ma...
Class XI Mathematics Mathematics 6 ( ) Hence n ! nC r , 0 ≤ r ≤ n. r !(n −r ) n n! n! 4. C n−r = = = nCr , (n −r)!(n− n( r ! ) (n − r)!r! Reprint 2026-27 PERMUTATIONS AND COMBINATION...
Class XI Mathematics Mathematics 6 the required number of ways = 3 3!2! 2 . Now, 1 man can be selected from 2 men in C ways 1nd 2 women can be selected from 3 women in C ways. Theref...
Class XI Mathematics Mathematics 6 spades.Hence,bymultiplicationprinciple,therequirednumberofways = C × 1 × C × 1 = 13 1 13 1 4 (iii) There are 12 face cards and 4 are to be selected...
Class XI Mathematics Mathematics 6 include exactly 4 bowlers? 8. A bag contains 5 black and 6 red balls. Determine the number of ways in which 2 black and 3 red balls can be selected...
Class XI Mathematics Mathematics 6 Therefore, the team can consist of (a) 1 boy and 4 girls (b) 2 boys and 3 girls (c) 3 boys and 2 girls (d) 4 boys and 1 girl. 7 4 1 boy and 4 girls...
Class XI Mathematics Mathematics 6 more than one digits given in the list is repeated, it will be A understood that in any number, the digits can be used as many times as is given in...
Class XI Mathematics Mathematics 6 and 9 which are divisible by 10 and no digit is repeated ? 6. The English alphabet has 5 vowels and 21 consonants. How many words with two differen...
Class XI Mathematics Mathematics 6 (n – 1) ! fi fi The number of permutations of n different things, taken r at a time, where repeatitionisallowed,isn .r fi The number of permutations o...
Class XI Mathematics Mathematics 6 instance, determined thenumberofcombinationsofknownplanetstakentwoatatime,threeatatime and so on. This was around 1140. It appears that Rabbi ben E...
Class XI Mathematics Mathematics 7 126 MATHEMATICS Chapter 7 BINOMIALTHEOREM v Mathematics is a most exact science and its conclusions are capable of absolute proofs. – C.P. STEINMET...
Class XI Mathematics Mathematics 7 row?Yes we do.Itcanbeseenthattheadditionof1’sintherowforindex1givesriseto2intherow for index 2. The addition of 1, 2 and 2, 1 in the row for index ...
Class XI Mathematics Mathematics 7 Observingthispattern,wecannowwritetherowofthePascal’striangleforanyindex without writing the earlier rows. For example, for the index 7 the row wou...
Class XI Mathematics Mathematics 7 i.e., k + 1 k + 1 k + 1 k + 1 k k + 1 k – 1 2 k + 1 k + 1 (a + b) = C a0 + C a1b + C 2 b + ...+ C k+1 b Now, (a + b) k + 1 = (a + b) (a + b) k k k ...
Class XI Mathematics Mathematics 7 P (k + 1) is true whenever P(k) is true. Therefore, by principle of mathematical induction, P(n) is true for every positive integer n. We illustrat...
Class XI Mathematics Mathematics 7 C x0+ C x + 1 x + ... 2 C x n 3 3 n n n Reprint 2026-27 BINOMIAL THEOREM 131 In particular, for x = 1, we have 2 = C + C0+ C + 1.. + C 2 n n (iii) ...
Class XI Mathematics Mathematics 7 + 5 × 100 × 16 – 32 =10040008000–1000800032=9039207968. Example 3 Which is larger (1.01) 1000000or 10,000? Solution Splitting 1.01 and using binomi...
Class XI Mathematics Mathematics 7 that when divided by 25, 6 – 5nleaves remainder 1. EXERCISE7.1 Expand each of the expressions in Exercises 1 to 5. 5  2 – x  6 1. (1–2x) 2.  x 2...
Class XI Mathematics Mathematics 7 The expansion of a binomial for any positive integral n is given by Binomial Theorem, which is (a + b) = C a + C a n 0 n n 1 n – b + C a 2 n – b + ...
Class XI Mathematics Mathematics 8 Chapter 8 SEQUENCESAND SERIES v Natural numbers are the product of human spirit. – DEDEKIND v 8.1 Introduction In mathematics, the word, “sequence”...
Class XI Mathematics Mathematics 8 SERIES 137 th In the sequence of primes 2,3,5,7,…, we find that there is no formula for the n prime. Such sequence can only be described by verbal ...
Class XI Mathematics Mathematics 8 9 and 11. n−3 1−3 1 1 (ii) Here a n . Thus, a1= = − ,a =2− ,a =0 3 4 4 2 4 Reprint 2026-27 138 MATHEMATICS 1 1 Hence, the first three terms are – ,...