how many different ways can the letters of \grammar\ be arranged?\nthe number of different ways that the…

Explanation:

Step1: Count total letters

The word "grammar" has 7 letters.

Step2: Identify repeated letters

The letter 'r' appears 3 times and the letter 'a' appears 2 times.

Step3: Use permutation formula for multi - sets

The formula for permutations of a multi - set with (n) objects, where (n_1,n_2,\cdots,n_k) are the number of times of repeated objects is (\frac{n!}{n_1!n_2!\cdots n_k!}). Here (n = 7), (n_1=3) (for 'r') and (n_2 = 2) (for 'a'). So the number of arrangements is (\frac{7!}{3!2!}).

Step4: Calculate factorial values

We know that (n!=n\times(n - 1)\times\cdots\times1). So (7!=7\times6\times5\times4\times3\times2\times1 = 5040), (3!=3\times2\times1=6) and (2!=2\times1 = 2).

Step5: Compute the result

(\frac{7!}{3!2!}=\frac{5040}{6\times2}=\frac{5040}{12}=420).

Answer:

420