Exams › JEE Advanced › Maths
The value of N = sumₖ₌₀¹⁰ C(30+k, k) * C(20-k, 10-k) = C(30,0)*C(20,10) + C(31,1)*C(19,9) + C(32,2)*C(18,8) +... + C(40,10)*C(10,0). Which of the following are INCORRECT?
- N is divisible by 10
- N is divisible by 3
- If N = C(x,y) for some x,y in positive integers, then (x+y) can equal 61
- N is divisible by 13
Correct answer: N is divisible by 13
Solution
By the Chu-Vandermonde identity, N = C(51,10) = 12777711870. This is divisible by 10 (ends in 0), divisible by 3 (digit sum 48), and equals C(51,10) so x+y=61 is achievable. It is NOT divisible by 13 (12777711870/13 is not an integer). So option D ('N is divisible by 13') is the INCORRECT statement.
Related JEE Advanced Maths questions
- If n is an odd number, how many ways can three terms in an arithmetic progression be chosen from the sequence 1, 2, 3,..., n?
- How many triangles can be created by connecting the vertices of a regular polygon with n (> 5) sides, ensuring that no triangle shares a side with the polygon?
- For n > 1, consider the product E = (2n + 1)(2n + 3)(2n + 5)...(4n - 3)(4n - 1). Which of the following statements is true?
- An individual has 6 friends and during a holiday period, he attended several dinners with them. He observed that he dined with all 6 friends on one occasion, with any group of 5 friends on 3 occasions, with any group of 4 friends on 3 occasions, with any group of 3 friends on 4 occasions, and with any pair of friends on 5 occasions. Additionally, each friend joined him for 7 dinners and missed 7 dinners. How many dinners did he attend alone?
- How many natural numbers n exist such that the factorial of n concludes with exactly 26 trailing zeros?
- In a dictionary arrangement, words are ordered starting with A, followed by C, E, K, L, and finally T. If there are 120 words for each of the first five starting letters (A, C, E, K, L), what will be the rank of the word 'TACKLE' when considering the words starting with T?
⚔️ Practice JEE Advanced Maths free + battle 1v1 →