Is it permissible for my friend to take the allowance deposited into her account after she withdrew from the college, especially since she cannot distinguish between what was deposited before her withdrawal and what was deposited after it?

If your friend did not take legal action to withdraw from the college, she is still officially a student and is entitled to the allowance, unless regulations or custom prevent it. However, if she has legally withdrawn, or if the regulations prohibit receiving the allowance after the decision to withdraw, then she is not permitted to take the allowances that were disbursed after the withdrawal date, and she must return them to the college administration. The allowances that were disbursed after her withdrawal can be identified by reviewing her bank statement.