Step-by-step Solution:

The word BANANA consists of 6 letters, with repeated letters. The distinct letters are ( B, A, N ), and their frequencies are as follows: \[\] ( A ) appears 3 times \[\] - ( N ) appears 2 times \[\] - ( B ) appears 1 time \[\] The formula to calculate the number of distinguishable permutations of a multiset is given by: \[\] Total permutations = \(\frac{n!} {n_1!} \cdot n_2! \cdot n_3! \) \[\] where: \[\] ( n ) = total number of items \[\] \( n_1, n_2, n_3 \) = frequencies of each repeated item \[\] For BANANA: \[\] - Total letters, \( n = 6 \) \[\] - Frequencies: \( n_1 = 3 \) (for ( A )), \( n_2 = 2 \) (for ( N )), \( n_3 = 1 \) (for ( B )) \[\] Substitute these into the formula: \[\] Total permutations \( \frac {6!}{3!} \cdot 2! \cdot 1! \) \[\] Step-by-step calculations:\[\] 1. ( 6! = 720 ) \[\] 2. ( 3! = 6 ) \[\] 3. ( 2! = 2 ) \[\] 4. ( 1! = 1 ) \[\] Now calculate the denominator:\[\] \( 3! \cdot 2! \cdot 1! = 6 \cdot 2 \cdot 1 = 12 \) \[\] Divide:\[\] Total permutations = \(\frac{720}{12} = 60 \) \[\] \[\] \[\] Correct Answer: (c)