Pulp solves multiple combinatorial problems

Question

As the title says,

I want to solve a problem similar to the summation of multiple schemes into a fixed constant, However, when I suggest the constrained optimization model, I can't get all the basic schemes well. Part of the opinion is to add a constraint when I get a solution. However, the added constraint leads to incomplete solution and no addition leads to a dead cycle.

Here is my problem description

I have a list of benchmark data detail_list ,My goal is to select several numbers from the benchmark data list(detail_list), but not all of them, so that the sum of these data can reach the sum of the number(plan_amount) I want.

For Examle

detail_list = [50, 100, 80, 40, 120, 25], plan_amount = 20, The feasible schemes are:

detail_list[2]=20 can be satisfied, detail_list[1](noly 10) + detail_list[3](only 10) = plan_amount(20) , detail_list[1](only 5) + detail_list[3](only 15) = plan_amount(20) also can be satisfied, and detail_list1 + detail_list2 + detail_list3 = plan_amount(20). But you can't take four elements in the detail_list are combined, because number = 3, indicating that a maximum of three elements are allowed to be combined.

from pulp import *
num = 6 #  the list max length
number_max = 3  # How many combinations can there be at most
plan_amount = 20
detail_list = [50, 100, 80, 40, 120, 25]    # Basic data
plan_model = LpProblem("plan_model")    
alpha = [LpVariable("alpha_{0}".format(i+1), cat="Binary") for i in range(num)]
upBound_num = [int(detail_list_money) for detail_list_money in detail_list]
num_channel = [
    LpVariable("fin_money_{0}".format(i+1), lowBound=0, upBound=upBound_num[i], cat="Integer") for i
    in range(num)]
plan_model += lpSum(num_channel) == plan_amount
plan_model += lpSum(alpha) <= number_max
for i in range(num):
    plan_model += num_channel[i] >= alpha[i] * 5
    plan_model += num_channel[i] <= alpha[i] * detail_list[i]
plan_model.writeLP("2222.lp")
test_dd = open("2222.txt", "w", encoding="utf-8")
i = 0
while True:
    plan_model.solve()
    if LpStatus[plan_model.status] == "Optimal":
        test_dd.write("第" + str(i + 1) + "次结果\n")
        for v in plan_model.variables():
            test_dd.write(v.name + "=" + str(v.varValue))
            test_dd.write("\n")
        test_dd.write("============================\n\n")
        alpha_0_num = 0
        alpha_1_num = 0
        for alpha_value in alpha:
            if value(alpha_value) == 0:
                alpha_0_num += 1
            if value(alpha_value) == 1:
                alpha_1_num += 1
        plan_model += (lpSum(
            alpha[k] for k in range(num) if value(alpha[k]) == 1)) <= alpha_1_num - 1
        plan_model.writeLP("2222.lp")
        i += 1
    else:
        break
test_dd.close()

I don't know how to change my constraints to achieve this goal. Can you help me