NCERT Class 11 Mathematics — Permutations and Combinations, Exercise 6.1. All 6 questions solved.

The whole exercise rests on the fundamental principle of counting:

If one task can be done in $m$ ways and, for each of those, a second task can be done in $n$ ways, then the two together can be done in $m \times n$ ways.

The technique that follows from it is to draw the slots and fill them one at a time, writing under each slot how many choices it has:

$$\underline{\ \ \ }\ \underline{\ \ \ }\ \underline{\ \ \ }$$

Whether the counts stay the same or drop by one at each step is decided by a single question: is repetition allowed?

Key insight. Fill the most constrained slot first. In question 2 the units digit must be even, so it has only $3$ choices while the others have $6$ — starting there keeps the constraint from interfering with the rest. This ordering matters far more in later exercises, where a slot filled in the wrong order changes the count.

Question 1

How many $3$-digit numbers can be formed from the digits $1$, $2$, $3$, $4$ and $5$ assuming that

(i) repetition of the digits is allowed?    (ii) repetition of the digits is not allowed?

Solution. There are three slots to fill — hundreds, tens and units — from a pool of five digits. Note that none of the five is $0$, so there is no leading-zero problem here.

(i) Repetition allowed. Each slot may take any of the five digits, independently of the others:

$$5 \times 5 \times 5 = 125$$

(ii) Repetition not allowed. The hundreds slot has $5$ choices; whichever digit is used is then unavailable, so the tens slot has $4$ and the units slot $3$:

$$5 \times 4 \times 3 = 60$$

Question 2

How many $3$-digit even numbers can be formed from the digits $1$, $2$, $3$, $4$, $5$, $6$ if the digits can be repeated?

Solution. A number is even exactly when its units digit is even, so that slot carries the constraint and should be filled first.

• Units: must be one of $2$, $4$, $6$ — that is $3$ choices.
• Hundreds: any of the six digits, since repetition is allowed — $6$ choices.
• Tens: likewise $6$ choices.

$$6 \times 6 \times 3 = 108$$

Question 3

How many $4$-letter codes can be formed using the first $10$ letters of the English alphabet, if no letter can be repeated?

Solution. The pool is $\{a, b, \ldots, j\}$, ten letters, and four slots to fill with no repetition:

$$10 \times 9 \times 8 \times 7 = 5040$$

Question 4

How many $5$-digit telephone numbers can be constructed using the digits $0$ to $9$ if each number starts with $67$ and no digit appears more than once?

Solution. The first two slots are fixed — they are $6$ and $7$, so they contribute a factor of $1$ each and nothing to count.

That leaves three slots and, since $6$ and $7$ are already used and no digit may repeat, a pool of $10 – 2 = 8$ remaining digits:

$$1 \times 1 \times 8 \times 7 \times 6 = 336$$

Question 5

A coin is tossed $3$ times and the outcomes are recorded. How many possible outcomes are there?

Solution. Each toss has two outcomes, head or tail, and the tosses are independent — a head on the first does not restrict the second. So

$$2 \times 2 \times 2 = 2^3 = 8$$

Listing them confirms the count: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.

Question 6

Given $5$ flags of different colours, how many different signals can be generated if each signal requires the use of $2$ flags, one below the other?

Solution. “One below the other” means the two positions are distinguishable — red above blue is a different signal from blue above red. So order matters, and the same flag cannot occupy both positions.

• Upper position: any of the $5$ flags.
• Lower position: any of the remaining $4$.

$$5 \times 4 = 20$$

Common mistakes

• Question 1(ii), using $5 \times 5 \times 5$ again. Once a digit is used it leaves the pool, so the counts fall $5, 4, 3$. Reading the phrase “repetition is not allowed” and then not acting on it is the single commonest error in the chapter.
• Question 2, applying the even condition to the wrong slot. A number is even according to its last digit, not its first. Filling the hundreds slot first with three choices gives $3 \times 6 \times 6$, which is the same number here by accident of arithmetic but the wrong reasoning — and it fails as soon as the digits are not all distinct in count.
• Question 2, forgetting that repetition is allowed. The units digit being $2$ does not stop the tens digit also being $2$; the answer would be $5 \times 4 \times 3$ if it did.
• Question 4, forgetting that $6$ and $7$ are used up. The remaining pool is eight digits, not ten. Answering $10 \times 9 \times 8 = 720$ is the result of taking the fixed prefix as costing nothing.
• Question 5, answering $6$. Three tosses of a two-sided coin is $2^3$, not $3 \times 2$. Slots multiply, they do not add.
• Question 6, dividing by $2$. The two flag positions are distinguishable, so order matters and no division is needed. Dividing by $2$ would be right for choosing two flags, which is a combination — the distinction the next exercises are built around.

Practise next

• Exercise 6.2 — factorial notation, which is the compact way of writing the falling products $5 \times 4 \times 3$ that appear throughout this exercise.
• Exercise 6.3 — permutations proper, where the slot-filling here is packaged as $^n\mathrm{P}_r$ and extended to repeated letters.