1
Q:

# If a+b+c =21 what is the total number of positive integral solutions?

 A) 109 B) 190 C) 901 D) 910

Explanation:

Number of positive integral solutions =  $\inline&space;_^{n-1}\textrm{C}_{r-1}$

= $\inline&space;^{21-1}\textrm{C}_{3-1}=^{20}\textrm{C}_{2}$

= 190

Q:

If two cards are taken one after another without replacing from a pack of 52 cards. What is the probability for the two cards be Ace ?

 A) 51/1221 B) 42/221 C) 1/221 D) 52/1245

Explanation:

Total Combination of getting a card from 52 cards = 52C1
Because there is no replacement, so number of cards after getting first card= 51
Now, Combination of getting an another card= 51C1
Total combination of getting 2 cards from 52 cards without replacement= (52C1)x(51C1)
There are total 4 Ace in stack. Combination of getting 1 Ace is = 4C1
Because there is no replacement, So number of cards after getting first Ace = 3
Combination of getting an another Ace = 3C1
Total Combination of getting 2 Ace without replacement=(4C1)x(3C1)
Now,Probability of getting 2 cards which are Ace = (4C1)x(3C1)/(52C1)x(51C1) = 1/221.

2 28
Q:

A Cricket team of 23 people all shake hands with each other exactly once. How many hand shakes occur ?

 A) 142 B) 175 C) 212 D) 253

Explanation:

The first person shakes hands with 22 different people, the second person also shakes hands with 22 different people, but one of those handshakes was counted in the 22 for the first person, so the second person actually shakes hands with 21 new people. The third person, 20 people, and so on...
So,
22 + 21 + 20 + 19 + 18 + 17 + 16 + 15 + 14 + 13 + 12 + 11 + 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1
= n(n+1)/2 = 22 x 23 /2 = 11 x 23 = 253.

2 32
Q:

There are three rooms in a Hotel: one single, one double and one for four persons. How many ways are there to house seven persons in these rooms ?

 A) 105 B) 7! x 6! C) 7!/5! D) 420

Explanation:

Choose 1 person for the single room & from the remaining choose 2 for the double room & from the remaining choose 4 people for the four person room,

Then, 7C1 x 6C2 x 4C4
= 7 x $\inline \fn_jvn \small \frac{6x5}{2}$ x 1
= 7 x 15 = 105.

4 63
Q:

How many parallelograms will be formed if 7 parallel horizontal lines intersect 6 parallel vertical lines?

 A) 215 B) 315 C) 415 D) 115

Explanation:
Parallelograms are formed when any two pairs of parallel lines (where each pair is not parallel to the other pair) intersect.
Hence, the given problem can be considered as selecting pairs of lines from the given 2 sets of parallel lines.
Therefore, the total number of parallelograms formed = 7C2 x 6C2 = 315

4 70
Q:

A polygon has 44 diagonals, then the number of its sides are ?

 A) 13 B) 9 C) 11 D) 7

Explanation:

Let the number of sides be n.

The number of diagonals is given by nC2 - n

Therefore , nC2 - n = 44, n>0

$\inline&space;\fn_jvn&space;\small&space;\frac{n(n-1)}{2}-n$ = 44
n- 3n - 88 = 0

n2 -11n + 8n - 88 = 0

n(n - 11) + 8(n - 11) = 0

n = -8 or n = 11.

As n>0, n will not be -8. Therefore, n=11.