Arranging Letters in TANZANIA with Restrictions

Understanding how to arrange the letters in the word ldquo;TANZANIArdquo; with specific restrictions is a fascinating exercise in combinatorial mathematics. This article will explore step-by-step how to calculate the number of ways to arrange these letters, with the constraint that the last letter cannot be Z.

Basic Combinatorial Formulas

First, let's establish the basic formulas for counting the arrangements of letters in a word. The formula for the number of distinct arrangements of letters is given by:

[ text{Total arrangements} frac{n!}{n_1! cdot n_2! cdot n_3! cdots} ]

where n is the total number of letters, and n_1, n_2, n_3, ldots> are the frequencies of the distinct letters.

Calculating Total Arrangements for TANZANIA

For the word ldquo;TANZANIArdquo;:

Total letters, n 8 A: 2 occurrences, n_A 2 N: 2 occurrences, n_N 2 Z: 2 occurrences, n_Z 2 T: 1 occurrence, n_T 1 I: 1 occurrence, n_I 1

Using the formula, the total number of distinct arrangements is:

[ text{Total arrangements} frac{8!}{2! cdot 2! cdot 2! cdot 1! cdot 1!} ]

Calculating the factorials:

8! 40,320 2! 2 1! 1

The total number of arrangements is then:

[ text{Total arrangements} frac{40,320}{2 cdot 2 cdot 2 cdot 1 cdot 1} frac{40,320}{8} 5,040 ]

Arrangements with Z at the End

Now, we need to subtract the arrangements where the last letter is Z. When Z is at the end, we have 7 remaining letters: T, A, N, Z, A, N, I. These 7 letters with repeated A’s and N’s can be arranged as follows:

[ text{Arrangements with Z last} frac{7!}{2! cdot 2! cdot 1! cdot 1!} frac{5,040}{4} 1,260 ]

Final Calculation

To find the number of ways to arrange the letters in ldquo;TANZANIArdquo; with the last letter not being Z, we subtract the arrangements with Z at the end from the total arrangements:

[ text{Arrangements where last letter is not Z} 5,040 - 1,260 3,780 ]

Alternative Methods

There are alternative methods to arrive at the same result. One approach is to first calculate the total arrangements before considering the restriction. Then, subtract the arrangements where Z is at the end:

Total arrangements of the 8 letters: [ frac{8!}{3! cdot 2!} 3,360 ] Arrangements with Z at the end: [ frac{7!}{3! cdot 2!} 420 ] Total ways to arrange the letters: [ 3,360 - 420 2,940 ]

Another method involves marking each A so it is unique and then calculating the permutations. This method yields:

[ 7 cdot 7! / 3! 77,654 / 6 491,200 - 1,200 58,800 - 1,200 58,680 ]

However, the most straightforward calculation confirms the correct number of arrangements is 3,780.

Conclusion

In conclusion, the number of ways to arrange the letters in the word ldquo;TANZANIArdquo; with the restriction that the last letter cannot be Z is 3,780. This exercise demonstrates the power of combinatorial mathematics in solving complex arrangement problems with specific constraints.