The number of terms in the expansion of $\left(y^{1/5} + x^{1/10}\right)^{55}$, in which powers of $x$ and $y$ are free from radical signs are
(1) six
(2) twelve
(3) seven
(4) five

The sum of the rational terms in the binomial expansion of $\left( 2 ^ { \frac { 1 } { 2 } } + 3 ^ { \frac { 1 } { 5 } } \right) ^ { 10 }$ is :
(1) 25
(2) 32
(3) 9
(4) 41

The sum of coefficients of integral powers of $x$ in the binomial expansion of $(1 - 2\sqrt{x})^{50}$ is:
(1) $\frac{1}{2}(3^{50} + 1)$
(2) $\frac{1}{2}(3^{50})$
(3) $\frac{1}{2}(3^{50} - 1)$
(4) $\frac{1}{2}(2^{50} + 1)$

The total number of irrational terms in the binomial expansion of $\left( 7 ^ { \frac { 1 } { 5 } } - 3 ^ { \frac { 1 } { 10 } } \right) ^ { 60 }$ is
(1) 48
(2) 55
(3) 54
(4) 49

If the number of integral terms in the expansion of $\left( 3 ^ { \frac { 1 } { 2 } } + 5 ^ { \frac { 1 } { 8 } } \right) ^ { n }$ is exactly 33, then the least value of $n$ is
(1) 264
(2) 128
(3) 256
(4) 248

The sum of all those terms which are rational numbers in the expansion of $\left( 2 ^ { \frac { 1 } { 3 } } + 3 ^ { \frac { 1 } { 4 } } \right) ^ { 12 }$ is:
(1) 89
(2) 27
(3) 35
(4) 43

If $n$ is the number of irrational terms in the expansion of $\left( 3 ^ { 1 / 4 } + 5 ^ { 1 / 8 } \right) ^ { 60 }$, then $( n - 1 )$ is divisible by :
(1) 26
(2) 30
(3) 8
(4) 7

The sum of all rational terms in the expansion of $\left( 1 + 2 ^ { 1 / 2 } + 3 ^ { 1 / 2 } \right) ^ { 6 }$ is equal to

Let the coefficients of three consecutive terms $T _ { r } , T _ { r + 1 }$ and $T _ { r + 2 }$ in the binomial expansion of $( a + b ) ^ { 12 }$ be in a G.P. and let $p$ be the number of all possible values of $r$. Let $q$ be the sum of all rational terms in the binomial expansion of $( \sqrt [ 4 ] { 3 } + \sqrt [ 3 ] { 4 } ) ^ { 12 }$. Then $\mathrm { p } + \mathrm { q }$ is equal to :
(1) 283
(2) 287
(3) 295
(4) 299

The least value of $n$ for which the number of integral terms in the Binomial expansion of $(\sqrt[3]{7} + \sqrt[12]{11})^n$ is 183, is:
(1) 2184
(2) 2196
(3) 2148
(4) 2172