Answer
a)210
b)386
c)848
d)252
Work Step by Step
a) Strings containing exactly 4 $1's$ = $C$(10,4) = 210
b) Strings containing at most 4 $1's$ = $C$(10,4) + $C$(10,3) + $C$(10,2) + $C$(10,1) + $C$(10,0) = 210 + 120 + 45 + 10 + 1 = 386
c) Strings containing at least 4 $1's$ =Total strings - stings with atmost 4 $1's$ + strings with exactly 4 $1's$ = $2^{10}$ - 386 + 210 = 848
d) Strings with 5 $1's$ = $C$(10,5) = 252