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69. Let X represent the number of spade cards among the five cards drawn. Since the drawing of card is with replacement, the trials are Bernoulli trials.

In a well shuffled deck of 52 cards, there are 13 spade cards.

$\begin{array}{l}\Rightarrow p=\frac{13}{52}=\frac{1}{4}\\ \therefore q=1-\frac{1}{4}=\frac{3}{4}\end{array}$

X has a binomial distribution with n=5 and $p=\frac{1}{4}$

$={\mathrm{5C}}_x$
$\mathrm{}{\left(\frac{3}{4}\right)}^{5-x}{\left(\frac{1}{4}\right)}^x$

P (all five cards are spades) $=P\left(X=5\right)$

$={\mathrm{5C}}_5$
$\begin{array}{l}\mathrm{}{\left(\frac{3}{4}\right)}^0.{\left(\frac{1}{4}\right)}^5\\ =1.\frac{1}{1024}\\ =\frac{1}{1024}\end{array}$

(ii) P (only 3 cards are spades) $=P\left(X=3\right)$

$=5C_3$
$\begin{array}{l}\mathrm{}{\left(\frac{3}{4}\right)}^2.{\left(\frac{1}{4}\right)}^3\\ =10.\frac{9}{16}.\frac{1}{64}\\ =\frac{45}{512}\end{array}$

(ii) P (none is a spades) $=P\left(X=0\right)$

$={\mathrm{5C}}_0$
$\begin{array}{l}\mathrm{}{\left(\frac{3}{4}\right)}^5.{\left(\frac{1}{4}\right)}^0\\ =1.\frac{243}{1024}\\ =\frac{243}{1024}\end{array}$

To get into the Andhra Medical College for the admission to DM and M.Ch courses via NEET SS exam, candidates need to score a cutoff rank between 327 and 1996 under the General AI quota, considering the Andhra Medical College NEET SS cutoff 2024 for the Round 1 seat allotment results.

68. Let X denote the number of defective items in a sample of 10 items drawn successively. Since the drawing is done with replacement, the trials are Bernoulli trials.

$\begin{array}{l}\Rightarrow p=\frac{5}{100}=\frac{1}{20}\\ \therefore q=1-\frac{1}{20}=\frac{19}{20}\end{array}$

X has a binomial distribution with n=10 and $p=\frac{1}{20}$

$={\mathrm{10C}}_x$

$\mathrm{}{\left(\frac{19}{20}\right)}^{10-x}.{\left(\frac{1}{20}\right)}^x$

P (not more than 1 defective item) $=P\left(X\le 1\right)$

$=P\left(X=0\right)+P\left(X=1\right)\\ ={\mathrm{10C}}_0$

$\mathrm{}{\left(\frac{19}{20}\right)}^{10}+{\left(\frac{1}{20}\right)}^0+10C_1$
$\begin{array}{l}\mathrm{}{\left(\frac{19}{20}\right)}^9+{\left(\frac{1}{20}\right)}^1\\ ={\left(\frac{19}{20}\right)}^{10}+10{\left(\frac{19}{20}\right)}^9.\left(\frac{1}{20}\right)\\ ={\left(\frac{19}{20}\right)}^9.\left[\frac{19}{20}+\frac{10}{20}\right]\\ ={\left(\frac{19}{20}\right)}^9.\left(\frac{29}{20}\right)\\ =\left(\frac{29}{20}\right).{\left(\frac{19}{20}\right)}^9\end{array}$

Yes, NIU offers scholarhips to international students. These students can be considered for a merit or housing scholarship. The scholarship application can be made via the Office of International admissions through MyScholarships application where these scholarships are listed. 

Below mentioned are some of NIU scholarships for international students: 

• International Freshman Merit Scholarship
• International Transfer Merit Scholarship
• Housing Scholarship

The latest cutoff details for B.Pharm. at Annai Veilankanni's Pharmacy College are not publicly specified. However, the general eligibility is 50% aggregate in 10+2 with Physics, Chemistry, and Biology/Mathematics, while SC/ST candidates require 40%. Meeting the eligibility is necessary, and admission is usually based on merit prepared from qualifying exam marks.

Considering the Andhra Medical College NEET SS cutoff 2024, the needed rank to get into DM and M.Ch. specialisations for the General AI category candidates stood between 347 and 1996. Moreover, if we talk about the lowest cutoff rank then it has been released for DM in Endocrinology. To know more, candidates can click here:

Candidates can compare Khwaja Moinuddin Chishti and Deen Dayal Upadhyaya Gorakhpur University for its MBA cousre based on multiple components such as fees, package, etc., to understand which university offers better MBA course: 

NOTE:This information is sourced from the official website/ sanctioning body and is subject to change.

67. The repeated tosses of a pair of dice are Bernoulli trials. Let X denote the number of times of getting doublets in an experiment of throwing two dice simultaneously four times.

Probability of getting doublets in a single throw of the pair of dice is

$\begin{array}{l}p=\frac{6}{36}=\frac{1}{6}\\ \therefore q=1-p=1-\frac{1}{6}=\frac{5}{6}\end{array}$

Clearly, X has the binomial distribution with n=4, $p=\frac{1}{6}$ , and $q=\frac{5}{6}$

$\therefore P\left(X=x\right){=}^n{C}_x{q}^{n-x}{p}^x$ , where $x=0,1,2,3...n$

$=4C_x$
$\mathrm{}{\left(\frac{5}{6}\right)}^{4-x}.{\left(\frac{1}{6}\right)}^x\\ =4C_x$
$\begin{array}{l}\mathrm{}.{\frac{5}{6^4}}^{4-x}\end{array}$

$\therefore P$ (2 successes) $=P\left(X=2\right)$

$={\mathrm{4C}}_2$
$\begin{array}{l}\mathrm{}.{\frac{5}{6^4}}^{4-2}\\ =6.\frac{25}{1296}\\ =\frac{25}{216}\end{array}$

In order to make a choice between BSc courses offered by IHM Bangalore and CII Institute of Hospitality, students can compare the options based on significant factors such as fees and Shiksha rating. To help, presented below is a comparison between the BSc courses at IHM Bangalore and CII Institute of Hospitality based on fees and ratings:

Note: The abovementioned information is as per official sources. However, it is subject to change. Hence, it is indicative.

The final application deadline for the NIILM University BCom programme 2025 has not been officially announced yet. However, registrations are currently open for BCom. Interested students are encouraged to regularly check the university’s official website or contact the admissions office for the latest updates. Staying informed will help ensure they don’t miss important deadlines for the NIILM BCom admission process.

Both IHM Bangalore and the School of Hospitality and Tourism Studies, DY Patil are good options to pursue a BSc. However, in terms of affordability, IHM Bangalore is more affordable since the tuition fees to pursue a BSc at IHM Bangalore is INR 3.17 lakh, and the tuition fee to pursue a BSc at the School of Hospitality and Tourism Studies, DY Patil is INR 12 lakh.

Note: The fee mentioned above is as per the official sources. However, it is indicative and subject to change.

Yes, the admission process for the BCom programme at NIILM University for the 2025 academic year is underway. Candidates who have completed their Class 12 from a recognised board with the required eligibility can apply. It is advisable to keep checking the official university website for the latest application details and updates.

The seat intake for MBA at Khwaja Moinuddin Chishti University is around 60. The MBA seats are offered based on the qualifying entrance test or as per the University's eligibility criteria. Further, seats are also reserved for the students belonging to reserved groups like Scheduled Caste, Scheduled Tribe, Other Backward Classes, and Economically Weaker Sections (EWS).

Recruiters such as PlanetSpark, TCS, Accenture, Jaypee Hospital and Justdial visited the campus during IIMT University placement 2025.

The average package received during IIMT University placement 2025 was INR 6 LPA

More than 2500 offers were extended and 1,000 recruiters have visited the campus during the past placement drives held at IIMT University.

Yes, Indians with tight finances may find it expensive to enroll in undergraduate or graduate programs at the Massachusetts Institute of Technology. The annual tuition fees for UG are between INR 50 L and INR 52 L for undergraduate and INR 35 L and INR 1 Cr for graduates. This shows the university is highly expensive. Also, students must be ready for additional costs required for accommodation, transportation, food, insurance, personal expenses, and more. The tuition fee table for popular courses is attached below:

Yes, IIMT University is good in terms of placements. The key highlights of IIMT University placement 2025 are presented below:

Note: The overall the highest package offered during the past placement drives was INR 33.5 LPA.