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Question

Consider the following paragraph:

THE ABILITY TO REASON ACCURATELY IS VERY IMPORTANT, AS IS THE ABILITY TO COUNT. AS AN EXERCISE IN BOTH, LET US COUNT HOW MANY TIMES THE LETTER "E" OCCURS IN THIS PARAGRAPH. THE CORRECT COUNT IS ________.

Which option when put in the blank in the above paragraph will make the final sentence accurate?

The correct answer is

SIXTEEN

Paragraph Analysis

The question asks us to find which word, when placed in the blank in the final sentence, makes that sentence an accurate statement about the number of times the letter "E" appears in the entire paragraph. This is a type of self-referential puzzle.

Let's break down the task:

  1. Identify the full paragraph, including the word inserted into the blank.
  2. Count the total number of times the letter "E" appears in this full paragraph.
  3. The word inserted in the blank must represent this total count for the final sentence to be accurate.

Let $W$ be the word placed in the blank, and let $N$ be the number represented by $W$. The final sentence is "THE CORRECT COUNT IS $W$". For this sentence to be accurate, the total count of 'E's in the entire paragraph must be equal to $N$.

The total count of 'E's in the paragraph is the sum of the 'E's in the paragraph text *before* the blank plus the 'E's within the word $W$ itself. Let $C_{before}$ be the count of 'E's in the paragraph text before the blank, and $C(W)$ be the count of 'E's in the word $W$.

The condition for accuracy is: $N = C_{before} + C(W)$.

Option Evaluation

We are given options for the word to put in the blank. Let's consider the provided correct option, "SIXTEEN".

  • If the word is "SIXTEEN", the number represented is $N = 16$.
  • We need to count the number of 'E's in the word "SIXTEEN". The letters are S, I, X, T, E, E, N. There are 3 'E's in "SIXTEEN", so $C(\text{SIXTEEN}) = 3$.

For the final sentence "THE CORRECT COUNT IS SIXTEEN" to be accurate, the total number of 'E's in the entire paragraph (with "SIXTEEN" in the blank) must be 16.

Using our condition $N = C_{before} + C(W)$, we substitute the values for the word "SIXTEEN":

$\qquad 16 = C_{before} + 3$

To find the required number of 'E's in the paragraph before the blank ($C_{before}$), we rearrange the equation:

$\qquad C_{before} = 16 - 3$

$\qquad C_{before} = 13$

Count Conclusion

This calculation shows that if the number of times the letter "E" occurs in the paragraph *before* the blank ("THE ABILITY TO REASON ACCURATELY IS VERY IMPORTANT, AS IS THE ABILITY TO COUNT. AS AN EXERCISE IN BOTH, LET US COUNT HOW MANY TIMES THE LETTER "E" OCCURS IN THIS PARAGRAPH. THE CORRECT COUNT IS") is 13, then inserting the word "SIXTEEN" (which contains 3 'E's) into the blank will result in a total of $13 + 3 = 16$ 'E's in the entire paragraph.

Since the final sentence then reads "THE CORRECT COUNT IS SIXTEEN", and the total count is indeed 16, the sentence becomes accurate.

Therefore, the word "SIXTEEN" makes the final sentence accurate.

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Important Questions from Fundamental Principles of Counting

  1. How many words starting with letter D can be formed by taking all letters from word DELHI, so that the letters are not repeated?

  2. Consider the digits 3, 5, 7, 9. What is the number of 5-digit numbers formed by these digits in which each of these four digits appears?

  3. Determine the quantity of different signals that can be created by organizing at least 2 flags in sequence (one below the other) on a vertical staff, if 5 different flags are available?

  4. A batch comprises of 4 red and 7 green uniform boys. Figure out the number of methods can a squad of 5 unit be picked if the squad has (i) at least one red and one green uniform boy (ii) at least 3 red uniform boys?

  5. The number of non-negative integers less than 1000 that contain the digit 1 are:

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