The food item(s) that can satisfy at least half of the MDR for all the three nutrients using up to seven servings of that single item alone is

A fair die is rolled repeatedly. The probability that the cumulative sum is at least 17 in the third trial is

If \(m\) is a positive integer then the values of \(k\) for which \(6m + k\) cannot be a perfect square are

Let \(S = \{1, 2, \ldots, 180\}\). Define \(A\) as the set of all multiples of 4 in \(S\), \(B\) as the set of all multiples of 6 in \(S\), and \(C\) as the set of all multiples of 9 in \(S\). The number of elements in \(S\) that belong to exactly one of \(A, B, C\) is

If \(\log_{18} 24 = p\), then \(\log_{96} 108\) equals

The number of integer solutions \((x, y)\) of the inequality \(x^2 + y^2 \leq 10\) is

If \(a\), \(b\), and \(c\) are three prime numbers such that \(abc = 23(a+b+c)\), then the maximum possible value of \(a+b-c\) is

The number of ways in which coins of denominations of rupees 2, 5 and 10 can be combined to make a value of rupees 50 is

Positive reals \(x, y\) satisfy \(x \neq y\) and \(\frac{x^2 + y^2}{xy} = k\). If replacing \(x\) by \(x+y\) and \(y\) by \(|x-y|\) leaves the value of \(k\) unchanged, then \(k\) equals