Get the sum with cases

Question

I have the following list:

list = {{1, 2}, {2}, {3, 4, 1}, {5, 4}, {3, 3}, {a, b, c}, {e, f}, {g}, {}, {Sin[a], Cos[b]}};

I should use Cases function Cases[list, pattern -> expr] with patterns or rules to get the following sums:

(a) the sum of all subexpressions with the length 2. Result:

{3,9,6,e+f,Cos[b]+Sin[a]}

(b) as in (a), but only for Integer numbers. Result:

{3,9,6}

(c) the sum of all subexpressions with the length >= 2. Result:

{3,8,9,6,a+b+c,e+f,Cos[b]+Sin[a]}

(d) the sum of ALL subexpressions. Result:

{3,2,8,9,6,a+b+c,e+f,g,0,Cos[b]+Sin[a]}

PS. It's not a homework. I learn for a test. This is an exercise I don't understand. Thanks for your help!

Answer 1

Cases[list, x : {_, _} :> Plus @@ x]
(* {3, 9, 6, e + f, Cos[b] + Sin[a]} *)

Cases[list, x : {_Integer, _Integer} :> Plus @@ x]
(* {3, 9, 6} *)

Cases[list, x : {_, __} :> Plus @@ x]
(* {3, 8, 9, 6, a + b + c, e + f, Cos[b] + Sin[a]}*)

Cases[list, x : {___} :> Plus @@ x]
(* {3, 2, 8, 9, 6, a + b + c, e + f, g, 0, Cos[b] + Sin[a]} *)

Or

patterns = {{_, _}, {_Integer, _Integer}, {_, __}, {___}};

Cases[list, x : # :> Plus @@ x] & /@ patterns
(* {{3, 9, 6, e + f, Cos[b] + Sin[a]}, 
    {3, 9, 6}, 
    {3, 8, 9, 6, a + b + c, e + f, Cos[b] + Sin[a]}, 
    {3, 2, 8, 9, 6, a + b + c, e + f, g, 0, Cos[b] + Sin[a]}} *)

Answer 2

Repeated can be very helpful. Also you can use +x as a shorthand for Plus[x], and x can be bound to a sequence. _Integer is the pattern for an expression with head Integer. (For my description of heads please see Is there a summary of answers Head[] can give? and Why can't a string be formed by head String?) Therefore for (b):

Cases[list, {x : Repeated[_Integer, {2}]} :> +x]

{3, 9, 6}

To sum all sublists I would either use Total or Map Tr across the list:

Total[list, {2}]

Tr /@ list

{3, 2, 8, 9, 6, a + b + c, e + f, g, 0, Cos[b] + Sin[a]}   (* output from either *)