How many ways can 7 red flags 1 white flag 1 yellow flag and 2 blue flags be arranged on the pole?

To find the number of ways to arrange 7 red flags, 1 white flag, 1 yellow flag, and 2 blue flags, we can use the formula for permutations of a multiset. The total number of flags is 11 (7 red + 1 white + 1 yellow + 2 blue). The formula is:

[ \frac{11!}{7! \cdot 1! \cdot 1! \cdot 2!} ]

Calculating this gives:

[ \frac{39916800}{5040 \cdot 1 \cdot 1 \cdot 2} = \frac{39916800}{10080} = 3960 ]

Thus, there are 3,960 distinct ways to arrange the flags on the pole.