Answer:  The given expression is 4n^3+ 6n^2+4n+1 =  4n^3+2n^2+4n^2+4n+1

= 2n^2(2n+1) +(2n+1)^2   = (2n+1){2n^2+(2n+1)} =  (2n+1)(2n^2+2n+1).

That is the given expression can be expressed as a product of 2 factors and both of these  factors are

natural numbers for all natural(number) values of "n". As the expression is a product of two natural numbers  it is a composite number for all natural values of n.

Answer:

There are only two combinations in which 22 employees can be arranged in 3 shifts such that at no time there will  more than 8 employees in one shift. They are i) 8+8+6  and ii) 8+7+7.

That is in the first combination, a group of 8 employees will be arranged ,say in I shift,another group of      8 employees  in II shift and the remaininmg employees in the third shift

In the second combination a group of 8 employees will be kept,say  in I shift,another group of 7 employees will be kept in II shift and the remaining 7 employees in the III shift.Of course,you can put any group in I shift or II shift or III shift as per your convenience. . 

i) 8+8+6: Let us call the first group of 8 employees as A and the second group of 8 employees B and the third group of 7 employees as C. Now let these 3 groups of employees  be arranged in First ,second and third  shift in the order of A,B,C respectively.It can be ensured that each employee gets 3rd shift once in 3 weeks by rotating their shifts in a cyclic manner.That is in the First week A,B,C come in I,II,III shiftsrespectively and in the second week the order will be B,C,A and in the third week the order will be  C,A,B.In fourth week the order A,B,C gets repeated and for the fifth  week, the order will be again B,C,A and so on. As already stated earlier, you can put any group in any shift as per your convenience.However after putting them in shifts in a particular order ,you have to  follow the cyclic order so as to ensure that each group gets one I shift,one II shift and one III shift once in three weeks. The following table shows the shift pattern for the above combination for 12 weeks.It can be easily seen that each employee gets one I shift,one II shift and one III shift once in 3 weeks...

Week  I Shift     II shift    III Shift    I Shift    II Shift   III Shift            

1           A          B             C             D       E           F   

2           B          C            A              E      F            D

3            C         A            B              F      D            E

4           A          B             C             D       E           F   

5           B          C            A              E      F            D

6            C         A            B              F      D            E

7           A          B             C             D       E           F   

8           B          C            A              E      F            D

9            C         A            B              F      D            E

  10         A          B             C             D       E           F   

11           B          C            A              E      F            D

12           C         A            B              F      D            E

Now there are 6 ways you can select the order of the groups.They are  ABC, ACB, BAC,BCA,CAB,CBA That  is you can choose ABC to  be in I,II and III shift respectively and then rotate them in cyclic order asabove.Or you can choose ACB to be in I,II and III shifts and rotate them in shifts in cyclic manner.Thus you have 6 patterns to begin with.  

ii) 8+7+7: Let us call them D,E,F.You can arrange these three  groups in I,II and III shift respectively and rotate their shifts in a cyclic order of DEF,EFD,FDE for the first  3 weeks and repeat them for the next 3 weeks and the next 3 weeks and so on.Here again you have 6 ways to select the order of the groups.They are DEF,DFE,EFD,EDF,FE and FDE. Thus you have in all 6+6 =12 patterns in which you can prepare the shift schedule for the 22 employees.

10×10 - 3 = 97 97×9 - 6 = 867 867×8 - 9 =6927 So this is answer 6927

132 की जगह 150 होगा

Solution uploaded

654326

34

9.25

94464 is the right answer

Yes both statement are true. Last monday will be on 25 from 1st statement and on 26 from 2nd statement