2022 AMC 10 A
Complete problem set with solutions and individual problem pages
How many strings of length formed from the digits , , , , are there such that for each , at least of the digits are less than ? (For example, satisfies the condition because it contains at least digit less than , at least digits less than , at least digits less than , and at least digits less than . The string does not satisfy the condition because it does not contain at least digits less than .)
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Suppose . Then let's do the casework.
① If numbers are all distinct, .
② If one number is counted twice, .
③ If two numbers are counted twice, .
④ If one number is counted times, .
⑤If one number is counted times and the others are counted twice, .
⑥ If one number is counted times, .
⑦ If one number is counted times, only 1 possible way.
In total, .
