He has a lot of problems ___ (look after)the baby alone at home.
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He has a lot of problems ___ (look after)the baby alone at home.
He has a lot of problems ___ (look after)the baby alone at home.
He has a lot of problems ___ (look after)the baby alone at home.
考查固定搭配
have problems (in) doing sth 做某事有困难
所以要用ing
looking after ,此处in省略了
他在家独自带小孩子有很大的困难.
一个人在家带小孩子对他来说很困难.
looking after
他独自在家照顾婴儿时遇到了很多问题
have trouble in doing sth 做某事遇到困难
looking after
语法：
have problems (in) doing sth
== have difficulty (in) doing sth.
He has a lot of problems looking after the baby alone at home.
to look after
固定搭配有两种，have problem to do sth. 或者have problem with doing sth.
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答案为in looking after，have problems/difficulties in doing sth 为固定搭配，表示做某事有困难或问题，in 也可以省略
to look after
这里用不定式的主动表被动
He has a lot of problems _in looking after__ baby alone at home.
他独自一人在家照顾小孩有很大困难
这个空填looking after，在空格前面省略了介系词in，表示“他在家独自照看孩子很困难”，动名词作介系词in的宾语。希望我能帮助你解疑释惑。
如果您原来想填空是to look after记得哟to接的是未发生的事情
正确答案是looking after因为做着,才能发现有困难
当然这也是,搭配哟*^_^*~