Permutations & Combinations
Permutation and Combination
star_batch_jee_advanced_2025
Grade 11
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:
Step-by-Step Solution
Key Concept: The last digit of a sum depends only on the last digits of the summands, and powers of 7 have a periodic cycle of length 4.
We need to find the last digit of $(13^n - 3^n) + 7^n = 10k + 7^n$. The last digit depends only on $7^n$. Since the last digits of powers of 7 cycle as $7, 9, 3, 1, 7, 9, 3, 1, ...$ with period 4, we have $7^n$ has last digit 3 when $n \equiv 3 \pmod{4}$, which means $n = 4k + 3$ for some integer $k \geq 0$.
Correct Answer: 3