The Mathsbombe Competition

2022 edition. From the people behind the Alan Turing Cryptography Competition.
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Problem 4

SEAT dealership

The pay structure at my local car dealership has the following rules:

  • All employees join the dealership at grade 0 (assistant sales facilitator)
  • At the end of each year of employment, an employee may be promoted to grade $N$, where the integer $N$ is less than or equal to the number of full years they have spent with the company, and greater than or equal to their current grade.

The deputy branch manager (a grade 6 role) has just completed their sixth year at the company. How many different ways could the deputy branch manager have been promoted to their current position?

Mathsbombe Competition 2022 is organised by the The Department of Mathematics at The University of Manchester.
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