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To begin with, we will have to know that the least number of beads we can equip. According to the question, we can at least include two beads of one color. Therefore, if we can make a bracelet with 4 beads in total, the answer must be 2 beads of one color + 2 beads of one color since the question asks us to at least equip 2 beads of one color. Following this rule, I calculated the following answers.

If we have 4 beads in total: 2 + 2
If we have 5 beads in total: 2 + 3
If we have 6 beads in total: 2 + 4, 3 + 3
If we have 7 beads in total: 2 + 5, 3 + 4
If we have 8 beads in total: 2 + 6, 3 + 5, 4 + 4
If we have 9 beads in total: 2 + 7, 3 + 6, 4 + 5
If we have 10 beads in total: 2 + 8, 3 + 7, 4 + 6, 5 +5