Try this problem from American Invitational Mathematics Examination, AIME, 2019 based on Combinations
A game show offers a contestant three prizes A B and C each of which is worth a whole number of dollars from $1 to $9999 inclusive. The contestant wins the prizes by correctly guessing the price of each prize in the order A B and C. As a hint the digits of three prizes are given. On a particular day the digits given were 1,1,1,1,3,3,3. Find the total number of possible guesses for all three prizes consistent with the hint.
Combinations
Theory of equations
Polynomials
But try the problem first...
Answer: 420.
AIME I, 2009, Problem 9
Combinatorics by Brualdi .
First hint
Number of possible ordering of seven digits is$\frac{7!}{4!3!}$=35
Second hint
these 35 orderings correspond to 35 seven-digit numbers, and the digits of each number can be subdivided to represent a unique combination of guesses for A B and C. Thus, for a given ordering, the number of guesses it represents is the number of ways to subdivide the seven-digit number into three nonempty sequences, each with no more than four digits. These subdivisions have possible lengths 1/2/4,2/2/3,1/3/3, and their permutations. The first subdivision can be ordered in 6 ways and the second and third in three ways each, for a total of 12 possible subdivisions.
Final Step
then total number of guesses is 35.12=420
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