A teacher bought a certain number of friendship bands for the students in the classroom for Friendship Day celebrations. Each student gives a friendship band to every other student. However, on the Friendship Day, 2 students were absent as a result of which 122 friendship bands were not utilized. But one out of the two absent students came late. Find the number of friendship bands which finally remained unutilized.

### Concept & Formula The total number of directional exchanges in a group (where everyone gives an item to everyone else) is calculated using permutations. $$Total = n(n-1)$$ ### Step-by-Step Solution * Let the total students in the class be $n$. * Total friendship bands initially required = $n(n - 1)$. * If 2 students are absent, the active students are $(n - 2)$. Bands utilized = $(n - 2)(n - 3)$. * Bands unutilized = $n(n - 1) - (n - 2)(n - 3) = 122$. * Expand the expressions: $n^2 - n - (n^2 - 5n + 6) = 122$. * Simplify the equation: $$4n - 6 = 122$$ * Solving for $n$: $4n = 128 \Rightarrow n = 32$. * If 1 of the absent students arrives late, only 1 student is ultimately absent. The active students = 31. * Unutilized bands = Total bought - New utilized = $32(31) - 31(30) = 31(32 - 30) = 31(2) = 62$. ### Exam Strategy & Shortcut If 2 students are missing, they don't give bands to the other $(n-2)$ students, and don't receive from them. Unutilized bands $= 2(n-1) + 2(n-2) = 4n - 6 = 122$. This quickly yields $n=32$. If only 1 student is absent, the unutilized bands simply equal $2(n-1) = 2(31) = 62$. ### Common Pitfall Students often confuse the formula for mutual exchanges $n(n-1)$ with handshakes $\frac{n(n-1)}{2}$. Friendship bands are directional (A gives to B, B gives to A), so dividing by 2 is conceptually incorrect here. ### Final Answer Therefore, the correct answer is 62.