Question

If all the arrangements formed using aU the letters of the following words are arranged In the order of dictionary then what will be the rank of that word?

(1) NIRAL

(2) SUMAN

Answer 1

(1) NIRAL

There are 5 letters N, I, R, A, L in the word NIRAL.

These 5 letters can be arranged in ⁵P₅ = 5! = 120 ways.

Now, we have to obtain the order of the word NIRAL from all 120 arrangements as per the dictionary order.

The alphabetical order of all letters of the word NIRAL is A, I, L, N, R.

Number of words with A at the first place

= ¹P₁ × ⁴P₄ = 1 × 4! = 24

Number of words with I at the first place

= ¹P₁ × ⁴P₄ = 1 × 4! = 24

Number of words with L at the first place

= ¹P₁ × ⁴P₄ = 1 × 4! = 24

Number of words with N at the first place and A at the second place

= ¹P₁ ×¹P₁ × ³P₃

= 1 × 1 × 3! = 1 × 6 = 6.

Number of words with N at the first place, I at the second place and A at the third place = ¹P₁ × ¹P₁ × ¹P₁ × ²P₂ = 1 × 2! = 2

Number of words with N at the first place, I at the second place and L at the third place = ¹P₁ × ¹P₁ × ¹P₁ × ²P₂ = 1 × 2! = 2

Now, the words with N at the first place, I at the second place, and R at the third place are NIRAL, NIRLA,

∴ Dictionary order of the word NIRAL = 24+ 24+ 24 + 6 + 2 + 2 + 1 = 83

(2) SUMAN

There are 5 letters S, U, M, A, N in the word SUMAN.

These 5 letters can be arrange in ⁵P₅ = 5! = 120 ways.

Now, we have to obtain the order of the word SUMAN from all 120 arrangements as per the dictionary order.

The alphabetical order of all letters of the word SUMAN is A, M, N, S, U.

Number of words with A at the first place

= ¹P₁ × ⁴P₄ = 1 × 4! = 24

Number of words with M at the first place

= ¹P₁ × ⁴P₄ = 1 × 4! = 24

Number of words with N at the first place

= ¹P₁ × ⁴P₄ = 1 × 4! = 24

Number of words with S at the first place and A at the second place

= ¹P₁ × ³P₃ = 1 × 3! = 6

Number of words with S at the first place and M at the second place

= ¹P₁ × ³P₃ = 1 × 3! = 6

Number of words with S at the place and N at the second place

= ¹P₁ × ³P₃ = 1 × 3! = 6

Number of words with S at the first place, U at the second place and A at the third place = ¹P₁ × ¹P₁ × ¹P₁ × ²P₂ = 1 × 2! = 2

Now, the words with S at the first place, U at the second place and M at the third place are SUMAN and SUMNA.

∴ Dictionary order of the word SUMAN is = 24+ 24+ 24 + 6 + 6 + 6 + 2+1 = 93