NCERT Solutions Ganita Prakash (Part 1) Chapter 2 .2 Exponential Notation and Operations · How Many Combinations — In-text Questions
Book page 262 Updated on2026-09-05
Q1.
Estu has 4 dresses and 3 caps. How many different ways can Estu combine the dresses and caps?
Answer
12 ways.
Counting by caps: for each of the 3 caps there are 4 dresses 4 + 4 + 4 = 4 × 3 = 12
Counting by dresses: for each of the 4 dresses there are 3 caps 3 + 3 + 3 + 3 = 3 × 4 = 12
Why it happens: the two counts agree because they count the same 12 pairs, once row-wise and once column-wise. This is the multiplication principle: when a choice is made in stages and the stages do not affect one another, multiply the numbers of options. It is the idea the whole password section rests on.
Every cap joins to every dress, so the lines number 3 × 4 = 12 — one line per outfit.
Q2.
Roxie has 7 dresses, 2 hats, and 3 pairs of shoes. How many different ways can Roxie dress up? [Hint: Try drawing a diagram like the one above.]
Answer
7 × 2 × 3 = 42 ways.
Choose a dress: 7 ways For each of those, choose a hat: 2 ways → 7 × 2 = 14 dress-and-hat pairs For each of those 14, choose shoes: 3 ways → 14 × 3 = 42
Why it happens: add a third stage and you multiply once more. Each of the 14 dress-and-hat choices splits into 3 branches for the shoes, so the count triples. Building the answer stage by stage is what makes the rule reliable — you never have to picture all 42 outfits at once.
Tip: the order of the stages does not matter: 2 × 3 × 7 and 3 × 7 × 2 also give 42. Choose whichever order makes the arithmetic easiest.
Q3.
Estu and Roxie came across a safe containing old stamps and coins that their great-grandfather had collected. It was secured with a 5-digit password. Since nobody knew the password, they had no option except to try every password until it opened. They were unlucky and the lock only opened with the last password, after they had tried all possible combinations. How many passwords did they end up checking?
Answer
105 = 1,00,000 passwords (one lakh).
Follow the book's advice and shrink the problem first.
Lock
Reasoning
Passwords
1-digit
0 to 9
10 = 101
2-digit
10 first digits × 10 second digits
100 = 102
3-digit
each of the 100 × 10 choices for the third
1000 = 103
5-digit
10 × 10 × 10 × 10 × 10
1,00,000 = 105
These are exactly the numbers 00000, 00001, 00002, …, 99998, 99999 — every whole number up to 99,999 written with five digits.
Why it happens: each extra slot multiplies the count by 10, because it multiplies every existing password into 10 new ones. The exponent in 105 is therefore the number of slots, not the number of digits available. This is why adding one character to a password makes it ten times harder to break, while adding one to its value makes no difference at all.
Did you know? At one password per second it would take 1,00,000 seconds — a little over a day of non-stop trying. Estu's worry about losing the whole vacation was not far off.