All of them are the same meaning, but I think in few situations, should only use one of them.
合わせる vs 併せる: あわせる has a lot of meanings, but roughly speaking, 合わせる is the default kanji that is safe in most situations. Basically there is no situation where you "have to" choose 併せる, but 併せる has an added nuance. When the kanji 併 is intentionally chosen, it tends to imply two things coexist side by side, whereas 合わせる often means two things are mixed, summed or merged into one. For example, 合わせて考える tends to look like "to consider taking everything into account", whereas 併せて考える tends to look like "to consider it along with others". As an exception, merger of organizations/municipalities is commonly described using 併.
組み合わせる: This is used when individual parts are unchanged but the resulting combination has some new meaning. Think of LEGO blocks or kanji compounds. 合わせる has many meanings, but 組み合わせる does not mean "to compare", "to match", etc.