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 contain not more than one letter is (K)6!. The value of K equals

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 contain not more than one letter is (K)6!. The value of K equals (A) 13 (B) 12 (C) 11 (D) 10

Tardigrade Admin · Accepted Answer

Any six blocks can be selected for 6 letters in

^{8} C_{6}

but this includes, when top 2 or middle two are not selected

∴

effective number of ways

=^{8} C_{6} - 2 =^{8} C_{2} - 2 = 26

six letters of "NATION" can be arranged in

\frac{6 !}{2}

ways
Total ways

= 26 \cdot \frac{6 !}{2} = 13 \cdot 6! \Rightarrow K = 13

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Q. 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 contain not more than one letter is (K)6!. The value of K equals

13

12

11

10

Any six blocks can be selected for 6 letters in 8C6​ but this includes, when top 2 or middle two are not selected ∴ effective number of ways =8C6​−2=8C2​−2=26 six letters of "NATION" can be arranged in 26!​ ways Total ways =26⋅26!​=13⋅6!⇒K=13.

Q. 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 contain not more than one letter is $(K) 6!$ . The value of K equals