Question

Use dynamic programming algorithm to compute the binary coefficient C(7,5). Show all steps!

Answer 1

in DP for binary coefficient ,

Recurrence relaton is:

C(n,k) = C(n-1,k) + C(n-1,k-1) for n > k > 0

                      C(n,0) = 1,   C(n,n) = 1 for n >= 0

we will make table, value of C(7,5) will be found from table

C(7,5) = C(6,5)+C(6,4) because [C(n,k) = C(n-1,k) + C(n-1,k-1) ]

c(column-wise) n(row-wise)	0	1	2	3	4	5
0
1
2
3
4
5
6
7

Step 1:   C(n,0) = 1,   C(n,n) = 1 for n >= 0

c(column-wise) n(row-wise)	0	1	2	3	4	5
0	1
1	1	1
2	1		1
3	1			1
4	1				1
5	1					1
6	1
7	1

Step 2: C(n,k) = C(n-1,k) + C(n-1,k-1) for n > k > 0

c(column-wise) n(row-wise)	0	1	2	3	4	5
0	1
1	1	1
2	1	2	1
3	1	3	3	1
4	1	4	6	4	1
5	1	5	10	10	5	1
6	1	6	15	20	15	6
7	1	7	21	35	35	21

C(7,5) = C(6,5)+C(6,4) = 6+15 =21