Dictionary Rank of a Word | Permutations & Combinations

 PERMUTATIONS & COMBINATIONS Rank of the word or Dictionary order of the English words like COMPUTER, COLLEGE, SUCCESS, SOCCER, RAIN, FATHER, etc. Dictionary Rank of a Word Method of finding the Rank (Dictionary Order) of the word  “R A I N” Given word: R A I N Total letters = 4 Letters in alphabetical order: A, I, N, R No. of words formed starting with A = 3! = 6 No. of words formed starting with I = 3! = 6 No. of words formed starting with N = 3! = 6 After N there is R which is required R ----- Required A ---- Required I ---- Required N ---- Required RAIN ----- 1 word   RANK OF THE WORD “R A I N” A….. = 3! = 6 I……. = 3! = 6 N….. = 3! = 6 R…A…I…N = 1 word 6 6 6 1 TOTAL 19 Rank of “R A I N” is 19 Method of finding the Rank (Dictionary Order) of the word  “F A T H E R” Given word is :  "F A T H E R" In alphabetical order: A, E, F, H, R, T Words beginni

NCERT Solution Class 9 Ch-1 Number System

Class 9 Chapter - 1 Number System
Numbers are the backbone of mathematics. Without numbers mathematics looks impossible. So students should have complete knowledge of number system. Before start this chapter students should go through some key points which are very important for accelerating the things.

BEFORE START NCERT CHAPTER 1 CLASS IX, STUDENTS SHOULD LEARN THE FOLLOWINGS

  • Tables (2-20),

METHOD OF FINDING RATIONAL NUMBERS BETWEEN GIVEN TWO NUMBERS.

Let two numbers are 2 and 3

These numbers can be written as 2.0 and 3.0

Rational numbers between 2 and 3 are as follows

2.1,  2.2,  2.3,  2.4,  2.5,  2.6,  2.7,  2.8,  2.9 etc.

When two fractions are given then algorithm to find the rational numbers between two fractions

Algorithm

1) Make the denominator of both the fractions equal by taking their LCM

2) Increase the gap between the numerators (if required) by multiplying by any larger number say 5, 10 etc.

3) Write the numbers between the numerators with the same denominator.

Example

Let two given numbers are :   
   and      

LCM of 3 and 5 is = 15

Now make both the denominators equal to 15

\[ \frac{2}{3}\times \frac{5}{5}=\frac{10}{15}\times \frac{10}{10}=\frac{100}{150}\]

\[\frac{4}{5}\times \frac{3}{3}=\frac{12}{15}\times \frac{10}{10}=\frac{120}{150}\]

Rational numbers between 2/3 and 4/5

\[ \frac{101}{150},\: \frac{102}{150},\: \frac{103}{150},\: \frac{104}{150},\: \frac{105}{150},\: \frac{106}{150}\]

METHOD OF CONVERTING RATIONAL DECIMALS INTO RATIONAL NUMBER

Example 1
\[ Let\: given\: number\: is=\: 2.\overline{4}\]
\[Let\: x=2.\overline{4}=2.444\: \: ....\: \: (1)\]

Multiplying on both side by 10 we get

10x = 10 X 2.444

10x = 24.444 .......... (2)

From Eqn. (2) - eqn. (1)

9x = 22  x = 22/9

Example 2
\[ Let\: given\: number\: is=\: 2.\overline{45}\]
\[Let\: x=2.\overline{4}=2.454545\: \: ....\: \: (1)\]

Multiplying on both side by 100 we get
100x = 100 X 2.4545 ...........

100x = 245.4545 .......... (2)

From Eqn. (2) - eqn. (1)

99x = 243  x = 243/99

Example 3
\[ Let\: given\: number\: is=\: 2.\overline{456}\]
\[Let\: x=2.\overline{4}=2.456456\: \: ....\: \: (1)\]
Multiplying on both side by 1000 we get

1000x = 1000 X 2.456456 ...........

1000x = 2456.456456 .......... (2)

From Eqn. (2) - eqn. (1)

999x = 2454  x = 2454/999

METHOD OF WRITING RATIONAL AND IRRATIONAL NUMBERS BETWEEN GIVEN TWO RATIONAL NUMBERS

Algorithm
1) Write the given rational number in the decimal form upto two decimal places.

2) Write the rational numbers between the given two numbers in the decimal form.

3) Convert the given rational numbers into irrationals by placing increasing order of number of zeroes between the digits.

Let given numbers are    and   

  =  0.7142 = 0.71 Approximately
 = 0.8181... = 0.81  Approximately

Irrational numbers between       and   

\[ 0.72,\: \:0.73,\: \: 0.75,\: \:0.76\: \: etc \]

Irrational numbers between    and  
\[ 0.72072007200072\: \: .........\]
\[ 0.73073007300073\: \: .........\]
\[ 0.75075007500075\: \: .........\]
\[ 0.76076007600076\: .......etc.\]

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