In a proof by induction of the claim:

For $n1,$ the number of $n-$bit strings that contain $n-1$ or more $1\text{'}$s is equal to $n+1.$

What would be the inductive step?

If the number of $k-$bit strings that contain $k-1$ or more $1\text{'}$s is equal to $k+1,$ then the number of $k+1-$bit strings that contain $k-1$ or more $1\text{'}$s is equal to $k+2.$

If the number of $k-$bit strings that contain $k-1$ or more $1\text{'}$s is equal to $k+1,$ then the number of $k+1-$bit strings that contain $k+1$ or more $1\text{'}$s is equal to $k+2.$

If the number of $k-$bit strings that contain $k-1$ or more $1\text{'}$s is equal to $k+1,$ then the number of $k+1-$bit strings that contain $k$ or more $1\text{'}$s is equal to $k+2.$

If the number of $k-$bit strings that contain $k-1$ or more $1\text{'}$s is equal to $k+1,$ then the number of $k+1-$bit strings that contain $k$ or more $1\text{'}$s is equal to $k+1.$