Question

If number of arrangements of letters of the word "DHARAMSHALA" taken all at a time so that no two alike letters appear together is (4^a.5^b.6^c.7^d), (where a, b, c, d ∈ N), then a + b + c + d is equal to ______.

Options

• 6.00

• 7.00

• 8.00

• 9.00

Answer 1

If number of arrangements of letters of the word "DHARAMSHALA" taken all at a time so that no two alike letters appear together is (4^a.5^b.6^c.7^d), (where a, b, c, d ∈ N), then a + b + c + d is equal to 7.00.

Explanation:

Required number of ways = Total when all A's separated – Total when A's separated and H's are together

= `(7!)/(2!)(""^8C_4) - 6!(""^7C_4)`

= `(7!6!)/(4!3!)(6)` = 4¹.5².6³.7¹

Now, 4^a.5^b.6^c.7^d = 4¹.5².6³.7¹ 

⇒ a + b + c + d = 1 + 2 + 3 + 1 = 7