Number of ways of selecting three integers from \( \{1,2,3 \), \( \ldots, n\} \) if their sum is...

Channel:
Subscribers:
449,000
Published on ● Video Link: https://www.youtube.com/watch?v=FFT8e1JjjMo



Duration: 2:52
9 views
1


Number of ways of selecting three integers from \( \{1,2,3 \), \( \ldots, n\} \) if their sum is divisible by 3 is
(1) \( 3\left({ }^{n / 3} C_{3}\right)+(n / 3)^{3} \) if \( n=3 k, k \in N \)
(2) \( 2\left({ }^{(n-1) / 3} C_{3}\right)+\left({ }^{(n+2) / 3} C_{3}\right)+((n-1) / 3)^{2}((n+2) / 3) \) if \( n=3 k+1, k \in N \)
(3) \( 2\left({ }^{(n-1) / 3} C_{3}\right)+\left({ }^{(n+2) / 3} C_{3}\right)+((n-1) / 3)^{2}((n+2) / 3) \) if \( n=3 k+2, k \in N \)
(4) independent of \( n \)
📲PW App Link - https://bit.ly/YTAI_PWAP
🌐PW Website - https://www.pw.live




Other Videos By PW Solutions


2022-10-27Differentiate the following w.r.t. \( x \) \[ e^{\cos ^{-1}\left(\sqrt{1-x^{2}}\right)},|x|1 \]
2022-10-27Differentiate the following functions w.r.t. \( x \). \[ x^{x} \]
2022-10-27Let a curve \( y=f(x) \) passes through \( (1,1) \), at any point \( P \) on the curve tangent a...
2022-10-27Let \( f(x)=\sin x, g(x)=x^{2} \) and \( h(x)=\log x \). If \( F(x)=h(f(g(x))) \), then \( F^{\p...
2022-10-27If \( 10 !=2^{p} \cdot 3^{q} \cdot 5^{r} \cdot 7^{s} \), then (1) \( 2 q=p \) (2) \( p q r s=64 ...
2022-10-27Let \( \alpha=\sin ^{-1}\left(\frac{36}{85}\right), \beta=\cos ^{-1}\left(\frac{4}{5}\right) \) ...
2022-10-27If \( a, b \), and \( c \) are in H.P., then the value of \( \frac{(a c+a b-b c)(a b+b c-a c)}{(...
2022-10-27Number of shortest ways in which we can reach from the point \( (0,0,0) \) to point \( (3,7,11) ...
2022-10-27Let \( S_{n}=\sum_{k=1}^{4 n}(-1)^{\frac{k(k+1)}{2}} k^{2} \). Then \( S_{n} \) can take value(s...
2022-10-27Given that the divisors of \( n=3^{p} \cdot 5^{q} \cdot 7^{r} \) are of the form \( 4 \lambda+1,...
2022-10-27Number of ways of selecting three integers from \( \{1,2,3 \), \( \ldots, n\} \) if their sum is...
2022-10-27The numbers \( a, b \), and \( c \) are between 2 and 18, such that (i) their sum is 25 (ii) the...
2022-10-27A circular ring of radius \( 3 \mathrm{~cm} \) hangs horizontally from a point \( 4 \mathrm{~cm}...
2022-10-27If \( \sin ^{-1}\left(a-\frac{a^{2}}{3}+\frac{a^{3}}{9}+\cdots\right)+\cos ^{-1}\left(1+b+b^{2}+...
2022-10-27If \( \cos ^{-1} x+\cos ^{-1} y+\cos ^{-1} z=\pi \), then (1) \( x^{2}+y^{2}+z^{2}+2 x y z=1 \) ...
2022-10-27If \( x0 \), then \( \tan ^{-1} x \) is equal to (1) \( -\pi+\cot ^{-1} \frac{1}{x} \) (2) \( \s...
2022-10-27Which of the following quantities is/are positive? (1) \( \cos \left(\tan ^{-1}(\tan 4)\right) \...
2022-10-27The number of real solutions of the equation \( \tan ^{-1} \sqrt{x^{2}-3 x+2}+\cos ^{-1} \sqrt{4...
2022-10-27The distance between the two parallel lines is 1 unit. A point ' \( A \) ' is chosen to lie betw...
2022-10-27If \( x^{2}+y^{2}+z^{2}=r^{2} \), then \( \tan ^{-1}\left(\frac{x y}{z r}\right)+\tan ^{-1}\left...
2022-10-27Which of the following is/are the value of \( \cos \left[\frac{1}{2} \cos ^{-1}\left(\cos \left(...



Tags:
pw