答案：C why.
翻译：那就是他们都反对这个计划的原因。
解释：the reason why/that的区别如下：
1. the reason why其后引导的是定语从句，且该定语从句缺少一个原因状语for the reason.
如此题：why they are all against the plan
= for which they are all against the plan
= they are all against the plan for which
= they are all against the plan for the reason
2. the reason that其后引导的也是定语从句，但是该定语从句不缺原因状语，只缺一个the reason做其主语/宾语/表语等名词形成分。
如：I don't believe the reason (that) you explained to me just now.
其中：(that) you explained to me just now
= you explained (that) to me just now
= you explained (the reason) to me just now
3. 不过在一些非正式场合中，the reason why也可以写成the reason that
原因：非正式用语中，关系代词that可以代替关系副词when/where/why，甚至可以省略不用。
4. 所以，如果此题没有选项C，则B也可以。
但是需要提醒楼主的是，此种用that代替when/where/why只能用于少数特殊句型或口语中，一般考试的时候，还是以关系副词为佳。