There are 5 bags labeled 1 to 5. All the coins in a given bag have the same weight. Some bags have coins of weight 10 gm, others have coins of weight 11 gm. I pick 1, 2, 4, 8, 16 coins respectively from bags 1 to 5. Their total weight comes out to 323 gm. Then the product of the labels of the bags having 11 gm coins is _____.

Let the number of coins picked from bags $1, 2, 3, 4, 5$ be $c_1 = 1, c_2 = 2, c_3 = 4, c_4 = 8, c_5 = 16$ respectively.

The total number of coins picked is:

If all coins had weighed $10\text{ gm}$, the total weight would be:

The actual measured weight is $323\text{ gm}$, so the excess weight is:

Since each coin of weight $11\text{ gm}$ contributes $(11 - 10) = 1\text{ gm}$ to the excess weight, the sum of coins taken from the bags containing $11\text{ gm}$ coins must equal $13$.

Expressing $13$ as a sum of powers of $2$ (its binary representation $13 = 1101_2$):

Thus, the bags containing $11\text{ gm}$ coins are Bag 1, Bag 3, and Bag 4.

The product of the labels of these bags is: