21

a)

5!

b)

4!

c)

5!

d)

4!

e)

3!

f)

0

23

Binomial[9, 5] 5! 8!

Binomial[9, 5]

2^13

Below is an exercise in Mathematica list manipulation to confirm the above calculation

All bit strings of length 13:

tt1 = PadLeft[#, 13] & /@ 
          Table[IntegerDigits[k, 2], {k, 0, 2^13 - 1}];

Length[tt1]

tt1[[8192]]

tt1[[234]]

Separate 0s and 1s by sorting:

Split[Sort[tt1[[123]]]]

Split[
      Sort[tt1[[8192]]]]

Split[Sort[tt1[[1]]]]

The last two bit strings are problematic so exclude them

And[(Length[Split[Sort[#]]] == 2), 
          Length[Split[
                  Sort[#]][[1]]] == 5] &[tt1[[123]]]

Select the bit strings with 5 0s:

tt2 = 
        Select[tt1, 
          And[(Length[Split[Sort[#]]] == 2), 
              Length[Split[
                      Sort[#]][[1]]] == 5] &];

Length[tt2]

Binomial[13, 5]

tt3 = Map[Split, tt2];

Length[tt3]

tt3[[123]]

Now somewhat complicated function to select only those strings that have no consecutive 0s:

MyF22[lll_] := 
      Apply[And, 
        Map[Or[And[(#[[1]] == 
                    0), Length[#] == 
                  1], (#[[1]] == 
                  1)] &, lll]]

MyF22[tt3[[123]]]

MyF22[tt3[[445]]]

tt4 = Select[tt3, MyF22[#] &];

Length[tt4]

Binomial[9, 5]

25

100 99 98 97

99 98 97

4 99 98 97

99 98 97 96

2! Binomial[4, 2] 98 97

3!  Binomial[4, 3] 97

4!

96 95 94 93

4 99 98 97

2 Binomial[4, 2] 96 95

29

(a)

98*97*4

98*97*4 - 97

(b)

98*72*2  -  97

30

(a)

Binomial[9, 5]

Binomial[16, 
      5]

Binomial[16, 5]  -  Binomial[9, 5]

Table[
      Binomial[7, k]*Binomial[9, 5 - k], {k, 1, 6}]

Sum[Binomial[7, k]*Binomial[9, 5 - k], {k, 1, 5}]

(b)

Sum[Binomial[7, k]*Binomial[9, 5 - k], {k, 1, 4}]

4242  -  Binomial[7, 5]

31

26^6

6 5 21^5

Binomial[6, 2] 5^2 21^4

26^6  -  21^6

26^6 - (21^6  +  
          6 5 21^5)

32

26^6

(a)

26^6  -  25^6

Sum[Binomial[6, k] 25^(6 - k), {k, 1, 6}]

(b)

24^6

26^6  -   24^6

2*(26^6  -  
            25^6)  - (26^6  -   24^6)

(c)

25 24 23 22 21

(d)

26 25 24 23 22 21

26 25 24 23 22 21 /2

33

Binomial[10, 3]*Binomial[15, 3]

34

Binomial[10 + 15, 6]

Binomial[10, 6]*Binomial[15, 0]

Binomial[10, 5]*Binomial[15, 1]

Binomial[10, 4]*Binomial[15, 2]

Binomial[10, 3]*Binomial[15, 3]

Binomial[10, 2]*Binomial[15, 4]

Binomial[10, 1]*Binomial[15, 5]

Binomial[10, 0]*Binomial[15, 6]

Sum[Binomial[10, k]*Binomial[15, 6 - k], {k, 0, 6}]

Binomial[10 + 15, 6]

Sum[Binomial[10, k]*Binomial[15, 6 - k], {k, 0, 2}]

35

Binomial[10, 8]

36

Binomial[14, 5]

37

2^10 - 2 (1 + 10 + Binomial[10, 2])

38

Binomial[45, 3] + Binomial[57, 4] + Binomial[69, 5]

How many possibilities are lost with the restrictions?

Binomial[45 + 57 + 69, 
        3 + 4 + 5] - (Binomial[45, 3] + Binomial[57, 4] + 
          Binomial[69, 5])

39

26*25*24*10*9*8

40

4!

{{a, b, c, d}, {b, c, d, a}, {c, d, a, b}, {d, a, b, c}}

{{a, b, d, c}, {b, d, c, a}, {d, c, a, b}, {c, a, b, d}}

6!/6

41

1  + 
   
      Binomial[3, 2]  +  
   
      Binomial[3, 1] ( Binomial[2, 2] ) + 
   3!

42

1  + 
   
      Binomial[4, 3]  +  
   
      Binomial[4, 2] (1  +  2)  + 
   
      Binomial[4, 1] ( 
          Binomial[3, 3]  +  Binomial[3, 2] + 
            Binomial[3, 1]) + 
   4!

43

Binomial[6, 6]  +  
      Binomial[6, 5] + Binomial[6, 4] + 
      Binomial[6, 3] + 
   
      Binomial[6, 
          2]*(1  +  Binomial[4, 3] + Binomial[4, 2] + 
            Binomial[4, 1]) + 
   
      Binomial[6, 1]*( 
          Binomial[5, 5]  +  Binomial[5, 4] + Binomial[5, 3] + 
            Binomial[5, 2]) + 
   
      Binomial[6, 1]*
        Binomial[5, 1]*( 
          Binomial[4, 4]  +  Binomial[4, 3] + Binomial[4, 2] + 
            Binomial[4, 1])