rac{7!}{3! \cdot 2! \cdot 2!}

February 22, 2026

$rac{7!}{3! \cdot 2! \cdot 2!}$

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Question: A science educator assigns 7 unique digital badgesâ3 for Robotics, 2 for Coding, and 2 for Data Scienceâto students over a week, one badge per day. If the badges within each category are indistinguishable, how many distinct daily sequences can be created?
Solution: We are arranging 7 badges where 3 are identical Robotics (R), 2 are identical Coding (C), and 2 are identical Data Science (D). The number of distinct sequences is the multinomial coefficient:
$ 7! = 5040 $
$ 3! = 6 $, $ 2! = 2 $, so denominator = $ 6 imes 2 imes 2 = 24 $
rac{5040}{24} = 210

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