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KVPY2017MathematicsBinomial Theorem

Among the inequalities below, which ones are true for all natural numbers n greater than 1000 ? I. n ! n ^ n II. (n !)² n^ n III. 10^ n n ! IV. n^ n (2 n ) !

Options

  1. AI and IV only
  2. BI, III and IV only
  3. CII and IV only
  4. DI, II, III and IV

Correct answer

B. I, III and IV only

Step-by-step solution

(A) n ^ n n ! = ( n n ) ( n n -1 ) ( n n -2 ) . ( n 1 ) 1 n ^ n n ! (correct) (C) n ! 10^ n = ( n )( n -1)( n -2) (10)(10) n times given that n >1000 clearly n ! 10^ n 1 n ! 10^ n (D) (1.2 .3 4 . . n)(n+1)(n+2)(n+3) (n+n) n^ n (n !) (1+ 1 n ) (1+ 2 n ) (1+ 3 n ) . (1+ n n ) clearly 1 . 2 n ! n^ n

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