In a certain algebraical exercise book there are 4 examples on arithmetical progressions, 5 examples on permutation and combination, and 6 examples on binomial theorem. Find the number of ways a teacher can select for his pupils at least one but not more than 2 examples from each of these sets.

Consider the number 2751. The sum of its digital is 2+7+5+1=15 . Adding the digits of 15 we get 1+5=6 . This number 6 is called the 'digital root' of the number 2751. That is, to find the digital root of a number, find the sum of its digits (Don't forget to find the sum of the digits again, if the first sum has more than one digit) Let us see one more example. The sum of the digits of the numbe 679412 is 6+7+9+4+1+2=29 . Sum of digits of 29=2+9=11 . Sum of digits of 11=1+1=2 . Therefore the digital root of 679412 is 2 . Digital roots have an interesting property. To see this, consider the product 43times27=1161 . The digital roots of the numbers 43 and 27 are 4+3=7 and 2+7=9 . Product of the digital roots= 7times9=63 . Digital root of 63=6+3=9 . The Digital root of 1161 is also 9(1+1+6+1=9) that is the digital root of 1161=The digital root of 63 , where 63 is the product of the digital roots of 43 and 27. This property is true for all other natural numbers. The digital root of the number 63square5 is 8 (square represents a missing digit). Find the missing digit.

Consider the number 2751. The sum of its digital is 2+7+5+1=15 . Adding the digits of 15 we get 1+5=6 . This number 6 is called the 'digital root' of the number 2751. That is, to find the digital root of a number, find the sum of its digits (Don't forget to find the sum of the digits again, if the first sum has more than one digit) Let us see one more example. The sum of the digits of the numbe 679412 is 6+7+9+4+1+2=29 . Sum of digits of 29=2+9=11 . Sum of digits of 11=1+1=2 . Therefore the digital root of 679412 is 2 . Digital roots have an interesting property. To see this, consider the product 43times27=1161 . The digital roots of the numbers 43 and 27 are 4+3=7 and 2+7=9 . Product of the digital roots= 7times9=63 . Digital root of 63=6+3=9 . The Digital root of 1161 is also 9(1+1+6+1=9) that is the digital root of 1161 =The digital root of 63 , where 63 is the product of the digital roots of 43 and 27. This property is true for all other natural numbers. If the digital root of a is 5 and the digital root of b is 2. Then what is the digital root of ab ?

Consider the number 2751. The sum of its digital is 2+7+5+1=15 . Adding the digits of 15 we get 1+5=6 . This number 6 is called the 'digital root' of the number 2751. That is, to find the digital root of a number, find the sum of its digits (Don't forget to find the sum of the digits again, if the first sum has more than one digit) Let us see one more example. The sum of the digits of the numbe 679412 is 6+7+9+4+1+2=29 . Sum of digits of 29=2+9=11 . Sum of digits of 11=1+1=2 . Therefore the digital root of 679412 is 2 . Digital roots have an interesting property. To see this, consider the product 43times27=1161 . The digital roots of the numbers 43 and 27 are 4+3=7 and 2+7=9 . Product of the digital roots= 7times9=63 . Digital root of 63=6+3=9 . The Digital root of 1161 is also 9(1+1+6+1=9) that is the digital root of 1161=The digital root of 63, where 63 is the product of the digital roots of 43 and 27. This property is true for all other natural numbers. 121times92=11square32 . Find the missing digit.

Consider the number 2751. The sum of its digital is 2+7+5+1=15 . Adding the digits of 15 we get 1+5=6 . This number 6 is called the 'digital root' of the number 2751. That is, to find the digital root of a number, find the sum of its digits (Don't forget to find the sum of the digits again, if the first sum has more than one digit) Let us see one more example. The sum of the digits of the numbe 679412 is 6+7+9+4+1+2=29 . Sum of digits of 29=2+9=11 . Sum of digits of 11=1+1=2 . Therefore the digital root of 679412 is 2 . Digital roots have an interesting property. To see this, consider the product 43times27=1161 . The digital roots of the numbers 43 and 27 are 4+3=7 and 2+7=9 . Product of the digital roots= 7times9=63 . Digital root of 63=6+3=9 . The Digital root of 1161 is also 9(1+1+6+1=9) that is the digital root of 1161=The digital root of 63 , where 63 is the product of the digital roots of 43 and 27. This property is true for all other natural numbers. What is the digital root of 345?

Consider the number 2751. The sum of its digital is 2+7+5+1=15 . Adding the digits of 15 we get 1+5=6 . This number 6 is called the 'digital root' of the number 2751. That is, to find the digital root of a number, find the sum of its digits (Don't forget to find the sum of the digits again, if the first sum has more than one digit) Let us see one more example. The sum of the digits of the numbe 679412 is 6+7+9+4+1+2=29 . Sum of digits of 29=2+9=11 . Sum of digits of 11=1+1=2 . Therefore the digital root of 679412 is 2 . Digital roots have an interesting property. To see this, consider the product 43times27=1161 . The digital roots of the numbers 43 and 27 are 4+3=7 and 2+7=9 . Product of the digital roots= 7times9=63 . Digital root of 63=6+3=9 . The Digital root of 1161 is also 9(1+1+6+1=9) that is the digital root of 1161=The digital root of 63 , where 63 is the product of the digital roots of 43 and 27. This property is true for all other natural numbers. What is the digital root of 927?

Consider the number 2751. The sum of its digital is 2+7+5+1=15 . Adding the digits of 15 we get 1+5=6 . This number 6 is called the 'digital root' of the number 2751. That is, to find the digital root of a number, find the sum of its digits (Don't forget to find the sum of the digits again, if the first sum has more than one digit) Let us see one more example. The sum of the digits of the numbe 679412 is 6+7+9+4+1+2=29 . Sum of digits of 29=2+9=11 . Sum of digits of 11=1+1=2 . Therefore the digital root of 679412 is 2 . Digital roots have an interesting property. To see this, consider the product 43times27=1161 . The digital roots of the numbers 43 and 27 are 4+3=7 and 2+7=9 . Product of the digital roots= 7times9=63 . Digital root of 63=6+3=9 . The Digital root of 1161 is also 9(1+1+6+1=9) that is the digital root of 1161 =The digital root of 63 , where 63 is the product of the digital roots of 43 and 27. This property is true for all other natural numbers. What is the digital root of 345times927 ?

A student has to answer 6 out of 10 questions which are divided into two parts containing 5 questions each and he is permitted to attempt not mom than 4 from any group. In how many ways can he make up his choice?

We know that sequence got by starting with any number and adding a fixed number repeatedly is called an arithmetic sequence.Example: 1,3,5,7,........ Like this we can form sequences by starting with any number and multiplying by a fixed non-zero number repeatedly. For example 1,2,4,8.......... In this sequence, one number multiplied by 2 gives the next number. Such sequences are called geometric sequences. The common number used for repeated multiplication is called common ratio. What is the 5^(th) term of the geometric sequence 1,2,4,8,........?