Question

Number of ways in which the letters of the word "NATION" can be filled in the given figure such that no row remains empty and each box contains not more than one letter, are:

• A. 116
• B. 126
• C. 136
• D. 146

Answer 1

The provided options are incorrect based on the standard interpretation of the problem.

Answer 2

The word "NATION" has 6 letters: N, A, T, I, O, N. The letter 'N' is repeated twice. The figure has 6 boxes arranged in three rows: Row 1 (2 boxes), Row 2 (1 box), and Row 3 (3 boxes).

The problem requires filling these 6 boxes with the 6 letters such that each box contains at most one letter and no row remains empty. Since there are 6 letters and 6 boxes, all boxes will be filled, automatically satisfying the "no row remains empty" condition.

The problem reduces to finding the number of distinct permutations of the letters of "NATION" in the 6 distinct boxes. The number of permutations of $n$ items where there are $n_1$ identical items of type 1, $n_2$ identical items of type 2, ..., $n_k$ identical items of type k is given by $\frac{n!}{n_1! n_2! \dots n_k!}$.

Here, $n=6$ (total letters), and the letters are N (2 times), A (1 time), T (1 time), I (1 time), O (1 time). So, the number of distinct permutations is: $\frac{6!}{2!1!1!1!1!} = \frac{6!}{2!}$ Calculating the factorials: $6! = 720$ $2! = 2$

Therefore, the number of ways is: $\frac{720}{2} = 360$ Since 360 is not among the given options (116, 126, 136, 146), there is likely an error in the question or the provided options.