This rule can be applied when we have two parameters whose product is constant, or in other words, when they are inversely proportional to each. Lets see the application with the following examples. Find the distance from his home to office? The power cycle of a number thus depends on its unit digit. Hence, we can say that 2 has a power cycle of 2,4,8,6 with frequency 4. Therefore to find the last digit of a number raised to any power, we just need to know matjs power cycle of digits from 0 to 9, which are given matjs.
Power cycle 0 1 2,4,8,6 3,9,7,1 4,6 5 6 7,9,3,1 8,4,2,6 9,1. For example 3 Find the remainder when is divided by 5. Hence the remainder is 2. Sometimes, you may get a question in the term of variables, where you need to substitute values to get the answer in the fastest way possible. For example. So taking the first digit from the power cycles of 3,8,2, and 7 we will get the unit digit as i.
We can classify the technique to be applied byhus 4 categories. In ; 2 is the tens digit of base and 7 is the unit digit of power. Change the power so that the base ends with 1 and then use the byjus maths questions 401 technique as for byjus maths questions 401 numbers ending with byjus maths questions 401. For byjus maths questions 401 last 2 digits of ,, 2.
What is the minimum possible number of men using all the 5 brands, if all the men use at least one of these brands? Similar to Different, which entails the number of ways of dividing n identical similar things into r distinct different groups.
Let me quesgions this with an example. Suppose I have 10 identical chocolates to be divided among 3 people. The 10 chocolates need to be distributed questilns 3 parts where a part can have zero questikns more chocolates. So let us represent chocolates by blue balls. The straight red byjus maths questions 401 are used to divide them into parts.
So you can see that for dividing into 3 parts, you need only two lines. Suppose you want to qiestions 1 st person 1 chocolate, 2nd 3 chocolates and 3rd 6 chocolates. Then you can show it as:. Suppose you questlons to give one person 1 chocolate, another person 6 chocolates and another one 3, then it can be represented as:.
Now if first person gets 0, second gets 1 and third gets 9 chocolates then it can be represented as:. Now suppose you want to byjus maths questions 401 first person 0, second also questionss and qiestions all of 10 then you can show it like:. So number of ways byjus maths questions 401 which you can questlons ten chocolates among 3 people is the same in which you can arrange 12 byjus maths questions 401 among which 10 are identical ybjus of one kind while 2 are identical and of one kind which can be done in 12!
In both the cases questiona answer is 12C2. Now suppose we have a restriction that the groups cannot be empty i. You have to divide ten chocolates among 3 persons so that each gets at least one. So in the start only give them one msths. This you will do in just 1 way as all the chocolates are identical. Now, you are left with 7 chocolates and you have to divide them among 3 people in such that way that Byjus Class 9 Maths Chapter 2 Raw each gets 0 or.
Byjus maths questions 401 can do this easily as explained above using balls and sticks. In both the cases the qkestions is 9C2.
Now suppose I change the question and say that now you have to divide 10 quesrions among 3 persons in such a way that one gets msths least 1, second at byjus maths questions 401 2 and third at least 3.
Its as simple as the last one. First full fill the required condition. Give 1 st person 1, second 2 and third 3 and then divide the left 4 chocolates among those 3 in such a way that each gets byjus maths questions 401 least 1. This is same as arranging 4 balls and 2 sticks byjus maths questions 401 can be done in 6C2 ways. In this case the answer is 9C2.
Rajesh went to the market to buy 18 fruits in all. Quextions there were mangoes, bananas, apples and oranges for sale then in how many byjus maths questions 401 matjs Rajesh buy at mats one fruit of each kind? This is a Grouping type 1 Similar to Distinct question, with a lower limit condition.
There are four balls to be put in five byjus maths questions 401 where each box can accommodate any number of balls. In how many ways can one do this if: a Balls are similar and boxes are different 1 2 70 c d By non-negative integral solutions, the conditions imply that we can have 0 and natural number values for x1, x2, x3, and x4 To remove the sign add another dummy questios x4.
Definition of Factorial N! A thorough understanding of Factorials is important because they play a pivotal role not only in understanding concepts in Numbers but also other important topics like Permutation and Combination. Questions based on highest power in a factorial are seen year after year in CAT. Questions based on this can be categorized based on the nature of the number prime or composite whose highest power byjus maths questions 401 are finding in the factorial, i.
To find the highest power of a prime number x in a factorial N! So highest power of 5 in ! Factorize the byjus maths questions 401 into primes. Find the highest power of all the prime numbers in that factorial using the previous uqestions. Take byjus maths questions 401 least power.
Highest power of 5 in ! Highest power of 2 in ! So take the lesser i. Find the highest power of 22 and 3 bgjus ! First find out the highest power of 2. Finding the number of zeroes forms the base concept for a number of application questions.
In base 10, Byjus Class 12 Maths Important Questions Pdf number of zeros in the end depends on the byjus maths questions 401 of 10s; i. Solution: We need to effectively find the highest power of 10 in 13! As this power will be lesser. Solution: Highest power of 12 in 15! STEP 1: Prime factorize 12! OR Find the remainder when 25! Both the questions are conceptually the same There are 6 zeroes at the end of 25!.
Effectively we need to find the byjus maths questions 401 non-zero integer in 25!. Number of 2s in 25! Answer is 4. If you know the unit digit in the product qestions This short cut method takes just 30 seconds.
Questiins this we can see that highest power of 5 till 10! Continuing like this, 10! Therefore, from 15! To 19! Therefore, 25! The answer to the question is therefore, 0.
There are no natural numbers whose factorials end with 5 zeroes. There are at least 10 questions out of 25 in CAT where you can apply this technique.
This involves assuming simple values for the variables in the questions, and substituting in answer options based on those values.
Assumption helps to tremendously speedup the process of evaluating the answer as shown. This is not direct substitution. In the next eg: see how you can use the same technique in an equation question. Define X as the average of odd integers in S and Y as the average of the even integers in S.
What is the value of X-Y? You can solve the question in less than 60 seconds. Check where you are getting 8 byjus maths questions 401 answer options. This Byjus Class 12 Maths Important Questions To Pdf wont take more than more than two minutes. Any two distinct members of S are called friends, if they have one constituent of the pairs in common and enemies. Here 1,21,3 are mathhs but 1,42,3 are enemies.
For general n, consider any two members of S that are friends. How many other members of S will be common friends of both these members? Then their common friends will be 1,4 questkons1,6 and 2,3.
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