| Class XI |
Mathematics |
Mathematics |
6 |
objects do not repeat is n ( n – 1) ( n – 2). . .( n – r + 1), which is denoted by P . r Proof There will be as many permutations as there are ways... |
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| Class XI |
Mathematics |
Mathematics |
6 |
2 × 1! Clearly, for a natural number n n ! = n (n – 1) ! = n (n – 1) (n – 2) ! [provided (n ≥ 2)] = n (n – 1) (n – 2) (n – 3) ! [provided (n ≥ 3)] ... |
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| Class XI |
Mathematics |
Mathematics |
6 |
n = 9, r = 5. 6.3.3 Derivation of the formula for Pn r nP = n! r (n−r !) , 0 ≤ r ≤ n Let us now go back to the stage where we had determined the fo... |
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| Class XI |
Mathematics |
Mathematics |
6 |
r 4 In Example 1, the required number of words = P = 4! = 44. Here repetition is notallowed. Ifrepetitionisallowed,therequirednumberofwordswouldbe4... |
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| Class XI |
Mathematics |
Mathematics |
6 |
O1R 2 TO O R T O O R 2 1 Reprint 2026-27 PERMUTATIONS AND COMBINATIONS 109 RO T1O 2 R O T O RO T 2 1 T O 1 O 2 T O R O T O R O 2 1 RT... |
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| Class XI |
Mathematics |
Mathematics |
6 |
U2T 2 Hen3e, total number of differentpermutationswillbe 9! 2!3! We can state (without proof) the following theorems: Theorem 3 The number of permu... |
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| Class XI |
Mathematics |
Mathematics |
6 |
taking 2 at a time. This number is P . So 6! 5! The required number = P −3P = 2 − 3! 3! = 4 × 5 × 6 – 4 ×5 = 100 Example 12 Find the value of n suc... |
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| 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... |
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| Class XI |
Mathematics |
Mathematics |
6 |
discs are 4 + 3 + 2 = 9. Out of 9 discs, 4 are of the first kind Reprint 2026-27 PERMUTATIONS AND COMBINATIONS 113 (red), 3 are of the second kind ... |
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| 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... |
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| 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... |
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| 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... |
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| 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... |
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| 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... |
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| 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... |
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| 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... |
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| 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... |
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| Class XI |
Mathematics |
Mathematics |
6 |
60 . To get the number of words starting withA, we fix the letterAat the extreme left position, we then rearrange the remaining 4 letters taken all... |
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| 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... |
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| 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... |
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| 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... |
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| Class XI |
Mathematics |
Mathematics |
6 |
instance, determined thenumberofcombinationsofknownplanetstakentwoatatime,threeatatime and so on. This was around 1140. It appears that Rabbi ben E... |
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| 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... |
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| 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 ... |
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| Class XI |
Mathematics |
Mathematics |
7 |
Observingthispattern,wecannowwritetherowofthePascal’striangleforanyindex without writing the earlier rows. For example, for the index 7 the row wou... |
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