A row of (2m + 2n) coins is arranged in a straight line, with (m + n) coins on each side of a central divider. The coins consist of 2m identical white coins and 2n identical red coins. In how many ways can the coins be arranged so that the arrangement is symmetric about the central divider

Question

A row of (2m + 2n) coins is arranged in a straight line, with (m + n) coins on each side of a central divider. The coins consist of 2m identical white coins and 2n identical red coins. In how many ways can the coins be arranged so that the arrangement is symmetric about the central divider?

Accepted Answer

C(m + n, m). Symmetry requires the right half to be the reflection of the left half. So we only arrange the left half: choose m positions (out of m+n) for white coins; the rest are red. This gives C(m+n, m) arrangements. Note C(m+n, m) = C(m+n, n), so both options A and B are correct and equivalent.

A row of (2m + 2n) coins is arranged in a straight line, with (m + n) coins on each side of a central divider. The coins consist of 2m identical white coins and 2n identical red coins. In how many ways can the coins be arranged so that the arrangement is symmetric about the central divider?

Solution

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