The total number of partitions of a k-element...

Q) The total number of partitions of a k-element
   set is denoted by B (k) and is called the k-th Bell number.

    Thus B (1) = 1and B (2) = 2.
   Need to find B (k) for k = 4, 5, 6.

Sol:

We know that,

Where,

Is the Binomial Coefficient.

It is given by:

It’s given that B (0) =1; B (1) =1; B (2) =2

Hence for n=3, from the first formulae

B (3) =C (2, K) B (K) where K=0 to n-1

B(3)=C(2,0)B(0)+ C(2,1)B(1)+ C(2,2)B(2)

From the second formulae, C (2, 0) =2! /0! (2-0) = 1

C (2, 1) =2; C (2, 2) =1

Hence,

B (3) =1*1+2*1+1*2=5

For n=4,

B(4)= C(3,0)B(0)+ C(3,1)B(1)+ C(3,2)B(2) + C(3,3)B(3)

        = 1*1+3*1+3*2+1*5= 15

Similarly,

For n=5,

B (5) = 52

For n=6,

B (6) = 203

Physics Homework: find the speed of the block after it has moved?

Q) 19 kg block initially at rest pulled to the right along a horizontal surface by a constant, horizontal force of 16.1 N. the coefficient of kinetic friction is 0.145. the acceleration of gravity is 9.8 m/s^2.find the speed of the block after it has moved 3.35 m.

Sol:

As per the concept that the work done by the force is equal to the Kinetic energy of the block, we have

u*(mg)*s=1/2mv² where work done= u*(mg)*s, u=0.145,m=19kg,g=9.8m/s², s=3.35m

Therefore by substituting the values in the above equation, we get,

0.145*9.8*3.35=0.5* v²

Hence Speed of the block after it has moved 3.35m is,

V= 3.086m/s