X, Y and Z are three non-negative integers such that X+Y+Z = 36. How many solutions are possible for this equation where \(X >= Y\) (i.e., X is greater than or equal to Y)?

(A) 319

(B) 341

(C) 361

(D) 385

(E) 405

EDIT:

(1) Corrected from "positve integers" to "non-negative integers".
(2) Added: X greater than or equal to Y.

Triangle ABC is a right angled triangle with right angle at Vertex B. Triangle DEF is an equilateral triangle with side 10 cm. The lengths of sides (in cm) are marked in the Drawing. What is the area of the polygon ABFEGA?

(1) The length of CD (overlap of sides) is 3 cm.
(2) The length of altitude (GH) of triangle GDC is 2.598 cm.





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This question can be done by two methods, first being conventional method, in which we will find conditional probabilities when the cards drawn in earlier draw is spade or not spade and adding all the cases.
But this method is tedious and lengthy.

My approach:
Since the number of cards of spade, heart, club and diamond is same.
The probability of each group of cards to appear in required draw will be same by symmetry. Let each probability be p.

Now since it is certain that the fifth card will be out of these four groups . TOTAL probability =1
p+p+p+p =1
Hence p =1/4.
Answer is 1/4.

Please provide Kudos if u like my approach.
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Q)How many different numbers can be formed from the products of numbers taken from set of 25 different prime numbers.(consider given is a set of first 25 prime numbers).

A. 2^25
B. 625
C. 25c2
D. 25p2
E. None of these






Please provide Kudos if you liked the questions.
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Only one of ten pens in a box is defective. Three pens are randomly drawn from the box, one at a time. What is the probability that the next pen randomly drawn is defective?

(A) 1/10
(B) 1/7
(C) 7/10
(D) 6/7
(E) 9/10

Source: Inspired from a Bunuel sir's question.
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Nice Question!

We can see that all the sides AB, BC, CD & DA are equal in length and forms a right angled triangle of sides 10 & 5 and each of AB, BC, CD & DA as hypotenuse
--> ABCD is a Rhombus as length of diagonal AC = \(10\sqrt{3}\) & BD = \(10\sqrt{2}\) are different

--> Area of Rhombus = 1/2*product of diagonlas = 1/2*\(10\sqrt{3}\)*\(10\sqrt{2}\) = \(50\sqrt{6}\)

IMO Option B

New Question

The square of the median of a sequence of consecutive positive odd integers is equal to the difference between the squares of the first and last terms. If the sequence has twelve terms, what is the sum of the digits of the first and last terms?

A. 12
B. 16
C. 20
D. 24
E. 28


source:self

The median of 12 consecutive odd integers is between the 6th and 7th terms, where if \(n > 1\), then
\(A_{n} = A_1 + 2(n-1)\)

\(A_1 = n\)
\(A_2 = n + 2\)
.
.
\(A_6 = n + 10\)
\(A_7 = n + 12\)
.
.
\(A_{12} = n + 22\)

Median = \(n + 11\). If not sure:

\(\frac{(n+10+n+12)}{2}=\frac{(2n+22)}{2}=(n + 11)\)

Square of median = difference between squares of first and last terms:

\((n + 22)^2 - (n)^2 = (n + 11)^2\)
\(n^2 + 44n + 484 - n^2 = n^2 + 22n + 121\)
\(n^2 - 22n - 363 = 0\)
\((n - 33)(n + 11) = 0\)
\(n = 33\) (prompt says positive)

\(A_1 = 33\)
\(A_{12} = n + 22 = 55\)

Sum of digits of \(33\) and \(55\):
\(3 + 3 + 5 + 5 = 16\)

ANSWER B
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Mr. White is an approximately forty years old father with 4 sons of distinct ages. Writing his age 3 times in succession, we get a 6-digit number that is equal to the product of his age, his wife's age and his 4 sons' ages. What is the sum of his wife's age and all 4 sons' ages?

A. 25
B. 30
C. 41
D. 51
E. 61

The shape of a window is a combination of semicircle and rectangle with semicircle QR on top and rectangle PQRS at the bottom. If arc QR is a semicircle and PQRS is a rectangle. What is the area of the window?

(1) The perimeter of rectangle PQRS is 28 feet
(2) Each diagonal of rectangle PQRS is 10 feet long



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For this we have to get 9! in prime factorization..

\(9!=1*2*3*4*5*6*7*8*9=2*3*2*2*5*2*3*7*2*2*2*3*3=2^7*3^4*5*7\)
Total factors = (7+1)(4+1)(1+1)(1+1)=8*5*2*2=160

Odd positive numbers will depend upon \(3^4*5*7\)....(4+1)(1+1)(1+1)=5*2*2=20

E

Hi
Very nice work.
I have framed this question myself. If you think, this is a good question, please provide Kudos...

\(\)
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=25c2+25c3+25c4+...25c25
=25c0+25c1+25c2+25c3+25c4+...25c25-(25c0+25c1)
=2^25 - 26
Answer is E.
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Another approach:
Let the prime numbers are p1,p2,...p25.
To find the product, each of the p1,p2... can be used or not, there is 2choices for each prime number.
2*2*2...2( 25 times) = 2^25
But this includes the cases when no number is selected (1 case) and the case when only one of the prime number is selected (25cases). These are unfavorable cases.
Hence answer is 2^25 -26. Hence answer is E
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Only Case 6 is favourable case

Area of shaded area = Area of square - 2 * Area of semicircle with diameter equal to side of square
= \(a^2\) - \(2*\frac{1}{2}*\pi*(\frac{a}{2})^2\)
= \(a^2\) -\(\pi*\frac{a^2}{4}\)
Total Area = Area of square =\(a^2\)
probability that both the triangles AED and BEC are acute angled triangle =\(\frac{Area of shaded area}{Total Area}\)
=\(\frac{(4-\pi)}{4}\)=0.215, Taking \(\pi\)=3.14
So OA=C

This question can be easily answered using ball parking.
See approach.


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No, you won't get such question on the tests. If a variable must be an integer then it will be an integer. So, the answer to your question is none of the options, it's flawed by GMAT standards.

P.S. Please post self-made questions here only: https://gmatclub.com/forum/users-self-m ... 54835.html Thank you.
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9! = 9x8x7x6x5x4x3x2x1
= \(2^7\) x \(3^4\)x 5 x 7

We know that the number of factors of a number will be expressed by the formula (p+1)(q+1)(r+1).....

To find odd factors/divisors, get rid of powers of 2 as they give even factors.

Odd factors/divisors of 9! = (4+1) x (1+1) x (1+1) = 20..........Option E

To find the odd factors we need to remove the factors of 2 present in 9!

All the prime factors below 9 are 2,3,5,7

And we can find the powers of each prime factor in 9!

we get that 9! = 2^7 * 3^4 * 5^1 * 7 ^1.
when we remove all the factors of 2.
we are left with 3^4 * 5^1 * 7 ^1.

the total number of ODD factors will be - (4+1)(1+1)(1+1) = 5*2*2 = 20 factors.

IMO option E is the answer.

If you like my solution, do give kudos!