Fundamental principle of counting

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In this Class 11 Mathematics topic from Permutations and Combinations, students learn how to count the total number of possible outcomes when a task is completed in successive stages. The fundamental principle of counting explains that if one step can be performed in one way and the next in another, the combined task can be counted by multiplying the choices at each stage. Students apply this idea to arrangements, selections, decision-making situations, and introductory permutation problems.

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A meal has (6) starters, (8) main dishes, and (5) desserts. With (2) special main dishes only (3) desserts are available, while with the rest (5) are available. How many meal combinations are possible?

A password has the form letter-digit-letter-digit-letter-digit. The first letter is a vowel, all three letters are distinct, all three digits are distinct, and the last digit is even. How many passwords are possible?

An identity number has (2) distinct letters followed by (4) odd digits. Letters cannot repeat and odd digits may repeat. How many identity numbers can be formed?

There are (7) routes from (X) to (Y), (6) routes from (Y) to (Z), and (4) routes from (Z) to (W). If the first route from (Y) to (Z) is chosen, only (2) later routes are available. How many journeys are possible?

One book is to be selected from each of four subjects. Mathematics has (5), Physics has (4), Chemistry has (6), and Biology has (3) books. The first Mathematics book and the second Chemistry book cannot be selected together. How many selections are possible?

In a (5)-digit PIN, digits cannot repeat, the first digit cannot be (0), and the digit (7) must occur exactly once. How many such PINs can be formed?

In (6) periods of a day, (8) subjects are to be placed without repetition. The last period must be one of (3) lab subjects, and the first period cannot be either of two non-lab subjects. How many timetables are possible?

For a T-shirt, there are (5) sizes, (7) colors, and (4) designs. With black color only (2) designs are available, and with other colors (4) designs are available. How many T-shirt combinations are possible?

An access code contains (2) distinct letters from (6) available letters, (2) symbols from (5) symbols, and (1) odd digit from (5) odd digits. Symbols may repeat. How many codes are possible?

An online test has (9) chapters. (3) chapters have (5) sets and (6) chapters have (4) sets, then (2) modes and (2) languages are selected. How many test forms are possible?

The first (2) places of a string are filled by distinct letters from (5) letters, and the last (2) places by distinct digits from (1,2,3,4). The last two digits must contain at least one even digit. How many strings are possible?

In a journey from (P) to (Q), exactly one intermediate hub must be chosen. Via (H_1), there are (3) and (4) route choices, while via (H_2), there are (5) and (2) route choices. How many journeys are possible?

A username has (2) distinct lowercase letters followed by (3) digits. Both letters cannot be vowels, and the (3) digits must be in increasing order. How many usernames are possible?

A locker system has (4) keys, (6) rooms, and (5) boxes. The first key opens only (3) boxes, while the other (3) keys open all (5) boxes. How many valid selections are possible?

Three-digit even numbers are to be formed with no repeated digit. How many such numbers are possible?

One representative is to be chosen from each of three clubs. Debate has (5) boys and (4) girls, Science has (6) boys and (3) girls, and Art has (4) boys and (5) girls. How many ways are there to choose exactly (2) girls?

A vehicle plate has (2) letters followed by (4) digits. Letters may repeat, digits do not repeat, the first digit is not (0), and the last digit is odd. How many plates can be formed?

One question is to be selected from each of four sections. The sections contain (7,8,6,5) questions respectively. Section one has (2) starred and section three has (3) starred questions, and both selected questions cannot be starred. How many valid papers are possible?

A CAPTCHA has the form digit-letter-digit-letter-digit. Letters are chosen from (21) consonants with repetition, digits are distinct, and the middle digit is greater than the last digit. How many CAPTCHAs are possible?

A coupon selects one each from (4) soups, (3) breads, (5) curries, and (6) drinks. With (2) juice drinks only (3) curries are available, while with other drinks (5) curries are available. How many coupons are possible?

A (4)-digit code is formed from digits (1) to (9) with no repeated digit. In the first two digits the first must be smaller, and in the last two digits the third must be larger. How many codes are possible?

A password first has one symbol from (4) symbols, then one vowel from (5) vowels, then (2) distinct consonants from (21) consonants, and finally one digit. If the first symbol is a special symbol, the digit cannot be (0). How many passwords are possible?

A library card has (2) distinct letters from (6) letters, then one separator from (3) separators, and then (2) digits. The sum of the last two digits must be odd and digits may repeat. How many cards are possible?

A train journey goes from (A) to (D) via (B) and (C). On the outward journey, there are (4,5,6) train choices respectively, and on return the same train cannot be reused on any corresponding segment. In how many ways can the round trip be made?

A portal requires selecting a region among (4) regions. (3) regions have (3) schools each and one region has only (2) schools; every school has (5) sections and every section has (2) monitors. How many monitor selections are possible?

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Fundamental principle of counting

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