New chat PYQ Library Doubts Rank report

Sign in to see Recents

Your guest activity stays on this device

Sign in to save progress →

Permutations and Combinations question

2024 · 4 Apr · Shift 1 · Q34

Guest · filters and generic practice availableBrowsing as a guest · PYQ filters and generic practice are available. Sign in only for personalised features and saved progress.

PYQ Library
JEE Main
Mathematics
Permutations and Combinations
2024 · 4 Apr · Shift 1 · Q34

Permutations and Combinations question

2024 · 4 Apr · Shift 1 · Q34

JEE MainMathematicsPermutations and CombinationsMCQ+4 / −1

There are 5 points

P_1, P_2, P_3, P_4, P_5

on the side

A B

, excluding

A

and

B

, of a triangle

A B C

. Similarly there are 6 points

\mathrm{P}_6, \mathrm{P}_7, \ldots, \mathrm{P}_{11}

on the side

\mathrm{BC}

and 7 points

\mathrm{P}_{12}, \mathrm{P}_{13}, \ldots, \mathrm{P}_{18}

on the side

\mathrm{CA}

of the triangle. The number of triangles, that can be formed using the points

\mathrm{P}_1, \mathrm{P}_2, \ldots, \mathrm{P}_{18}

as vertices, is:

A
751
B
776
C
796
D
771

View written solutionFree

Correct answer: A

Total number of given points

There are:

$5$ points on side $AB$
$6$ points on side $BC$
$7$ points on side $CA$

So total points are $5+6+7=18.$

Total ways to choose any 3 points

If no three were collinear, the number of triangles would be $\binom{18}{3} = \frac{18\cdot 17\cdot 16}{6} = 816.$

Subtract collinear selections

A triangle is not formed when all 3 chosen points lie on the same side of the triangle.

On side $AB$ , there are $5$ points, so collinear triples: $\binom{5}{3}=10.$
On side $BC$ , there are $6$ points, so collinear triples: $\binom{6}{3}=20.$
On side $CA$ , there are $7$ points, so collinear triples: $\binom{7}{3}=35.$

Total collinear triples: $10+20+35=65.$

Number of triangles

Hence required number of triangles is

=816-65=751.$$ 5. **Option check** - A: $751$ ✅ - B: $776$ ❌ - C: $796$ ❌ - D: $771$ ❌ Therefore, the correct answer is **Option A**.

More from Permutations and Combinations

There are 4 men and 5 women in Group A, and 5 men and 4 women in Group B. If 4 persons are selected from each group, then the number of ways of selecting 4 men and 4 women is .2024 · Numerical
The number of ways of getting a sum 16 on throwing a dice four times is .2024 · Numerical
Let the set $S = {2, 4, 8, 16, \dots, 512}$ be partitioned into 3 sets $A, B, C$ with equal number of elements such that $A \cup B \cup C = S$ and $A \cap B = B \cap C = A \cap C = ϕ$ …2024 · MCQ
60 words can be made using all the letters of the word $BHBJO$ , with or without meaning. If these words are written as in a dictionary, then the $5 0^{th}$ word is:2024 · MCQ
The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is2024 · MCQ
If all the words with or without meaning made using all the letters of the word "NAGPUR" are arranged as in a dictionary, then the word at $31 5^{th}$ position in this arrangement is :2024 · MCQ
Let $0 \leq r \leq n$ . If $^{n + 1} C_{r + 1} :^{n} C_{r} :^{n - 1} C_{r - 1} = 55 : 35 : 21$ , then $2 n + 5 r$ is equal to :2024 · MCQ
Let $[t]$ be the greatest integer less than or equal to $t$ . Let $A$ be the set of all prime factors of 2310 and $f : A \to Z$ be the function $f (x) = [lo g_{2} (x^{2} + [\frac{x ^{3}}{5}])]$ . The…2024 · MCQ